978-3-98547-561-2-volume-3

INTERNATIONAL GONGRESS

\overset{Â¯}{F}

the group of its cocharacters.

For a prime

ℓ

, let

Λ

F_{ℓ}, Z_{ℓ}, Q_{ℓ}

or a finite (flat) extension of such rings. It will serve as the coefficient ring of our sheaf theory.

1. FROM CLASSICAL TO GEOMETRIC LANGLANDS CORRESPONDENCE

In this section, we review some developments of the geometric Langlands theory inspired from the classical theory, with another important source of inspiration from quantum physics. The basic idea is categorification/geometrization, which is a process of replacing set-theoretic statements with categorical analogues

\begin{matrix} (1.1) & Numbers \to Vector spaces \to Categories \to 2-Categories \to \dots . \end{matrix}

We illustrate this process by some important examples.

1.1. The geometric Satake

The starting point of the Langlands program is (Langlands' interpretation of) the Satake isomorphism, in which the Langlands dual group appears mysteriously. Similarly, the starting point of the geometric Langlands theory is the geometric Satake equivalence, which is a tensor equivalence between the category of perverse sheaves on the (spherical) local Hecke stack of a connected reductive group and the category of finite-dimensional algebraic representations of its dual group. This is a vast generalization of the classical Satake isomorphism (via the sheaf-to-function dictionary), and arguably gives a conceptual explanation why the Langlands dual group (in fact, the

C

-group) should appear in the Langlands correspondence.

We follow [83, SECT. 1.1] for notations and conventions regarding dual groups. Let

G

denote a connected reductive group over a field

F

. Let

(\hat{G}, \hat{B}, \hat{T}, \hat{e})

be a pinned Langlands dual group of

G

over

Z

. There is a finite Galois extension

\tilde{F} / F

, and a natural injective map

ξ : Γ_{\tilde{F} / F} \subset Aut (\hat{G}, \hat{B}, \hat{T}, \hat{e})

, induced by the action of

Γ_{F}

on the root datum of

G

. Let

^{L} G = \hat{G} ⋊ Γ_{\tilde{F} / F}

denote the usual

L

-group of

G

, and

^{c} G = \hat{G} ⋊ (G_{m} \times Γ_{\tilde{F} / F})

the group defined in [83], which is isomorphic to the

C

-group of

G

introduced by Buzzard-Gee. We write

d :^{c} G \to G_{m} \times Γ_{\tilde{F} / F}

for the projection with the kernel

\hat{G}

Now let

F

be a nonarchimedean local field with

O

being its ring of integers and

k = F_{q}

its residue field. That is,

F

is a finite extension of

Q_{p}

or is isomorphic to

F_{q} ((ϖ))

. Let

σ

be the geometric

q

-Frobenius of

k

. Assume that

G

can be extended to a connected reductive group over

O

(such

G

is called unramified), and we fix such an extension to have

G (O) \subset G (F)

, usually called a hyperspecial subgroup of

G (F)

. With a basis of open neighborhoods of the unit given by finite-index subgroups of

G (O)

, the group

G (F)

is a locally compact topological group. The classical spherical Hecke algebra is the space of compactly supported

G (O)

-biinvariant

C

-valued functions on

G (F)

, equipped with the convolution product

\begin{matrix} (1.2) & (f * g) (x) = \int_{G (F)} f (y) g (y^{- 1} x) d y \end{matrix}

where

d y

is the Haar measure on

G (F)

such that

G (O)

has volume 1 . Note that if both

f

and

g

are

Z

-valued, so is

f * g

. Therefore, the subset

H_{G (O)}^{c l}

Z

-valued functions forms a

Z

-algebra.

^{1}

On the dual side, under the unramifiedness assumption,

Γ_{\tilde{F} / F}

is a finite cyclic group generated by

σ

. Note that

\hat{G}

acts on

{^{c} G |}_{d = (q, σ)}

, the fiber of

d

(q, σ) \in G_{m} \times Γ_{\tilde{F} / F}

, by conjugation. Then the classical Satake isomorphism establishes a canonical isomorphism of

Z [q^{- 1}]

-algebras

\begin{matrix} (1.3) & {S a t}^{c l} : Z [q^{- 1}] {[{^{c} G |}_{d = (q, σ)}]}^{\hat{G}} ≅ H_{G (O)}^{c l} \otimes Z [q^{- 1}] \end{matrix}

Remark 1.1.1. In fact, as explained in [83], there is a Satake isomorphism over

Z

(without inverting

q

), in which the

C

-group

^{c} G

is replaced by certain affine monoid containing it as the group of invertible elements. On the other hand, if we extend the base ring from

Z [q^{- 1}]

Z [q^{\pm \frac{1}{2}}]

, one can rewrite (1.3) as an isomorphism

\begin{matrix} (1.4) & Z [q^{\pm \frac{1}{2}}] [\hat{G} σ]^{\hat{G}} ≅ H_{G (O)}^{c l} \otimes Z [q^{\pm \frac{1}{2}}] \end{matrix}

where

\hat{G}

acts on

\hat{G} σ \subset^{L} G

by the usual conjugation (e.g., see [83] for the discussion). This is the more traditional formulation of the Satake isomorphism, which is slightly noncanonical, but suffices for many applications.

In the geometric theory, where instead of thinking

G (F)

as a topological group and considering the space of

G (O)

-biinvariant compactly supported functions on it, one regards

1 Here

(-)^{c l}

stands for the classical Hecke algebra, as opposed to the derived Hecke algebra mentioned in (2.2).

G (F)

as a certain algebro-geometric object and studies the category of

G (O)

-biequivariant sheaves on it. In the rest of the section, we allow

F

to be slightly more general. Namely, we assume that

F

is a local field complete with respect to a discrete valuation, with ring of integers

O

and a perfect residue field

k

of characteristic

p > 0 .^{2}

Let

ϖ \in O

be a uniformizer.

We work in the realm of perfect algebraic geometry. Recall that a

k

-algebra

R

is called perfect if the Frobenius endomorphism

R \to R, r \mapsto r^{p}

, is a bijection. Let

{A f f}_{k}^{p f}

denote the category of perfect

k

-algebras. By a perfect presheaf (or, more generally, a perfect prestack), we mean a functor from

{A f f}_{k}^{p f}

to the category Sets of sets (or, more generally, a functor from

A f f_{k}^{p f}

to the

\infty

-category

S p c

of spaces). Many constructions in usual algebraic geometry work in this setting. For example, one can endow Aff

_{k}^{p f}

with Zariski, Ã©tale, or fpqc topology as usual and talk about corresponding sheaves and stacks. One can then define perfect schemes, perfect algebraic spaces, perfect algebraic stacks, etc., as sheaves (stacks) with certain properties. It turns out that the category of perfect schemes/algebraic spaces defined this way is equivalent to the category of perfect schemes/algebraic spaces in the usual sense. Some foundations of perfect algebraic geometry can be found in [78, APPENDIX A], [13] and [64, SECT. A.1].

For a perfect

k

-algebra

R

, let

W_{O} (R)

denote the ring of Witt vectors in

R

with coefficient in

O

. If char

F =

char

k

, then

W_{O} (R) ≃ R [[ϖ]]

. If char

F \neq

char

k

, see [78, SECT. 0.5]. If

R = \bar{k}

, we denote

W_{O} (\bar{k})

O_{\overset{˘}{F}}

and

W_{O} (\bar{k}) [1 / ϖ]

\overset{˘}{F}

. We write

D_{R} = Spec W_{O} (R)

and

D_{R}^{*} = Spec W_{O} (R) [1 / ϖ]

which are thought as a family of (punctured) discs parameterized by

Spec R

We denote by

L^{+} G

(resp.

L G

) the jet group (resp. loop group) of

G

. As presheaves on Aff

_{k}^{pf}

L^{+} G (R) = G (W_{O} (R)), L G (R) = G (W_{O} (R) [1 / ϖ])

Note that

L^{+} G (k) = G (O)

and

L G (k) = G (F)

. Let

{H k}_{G} := L^{+} G ∖ L G / L^{+} G

be the Ã©tale stack quotient of

L G

by the left and right

L^{+} G

-action, sometimes called the (spherical) local Hecke stack of

G

. As a perfect prestack, it sends

R

to triples

(E_{1}, E_{2}, β)

, where

ε_{1}, ε_{2}

are two

G

-torsors on

D_{R}

, and

β : {{ε_{1} |}_{D_{R}^{*}} ≃ ε_{2} |}_{D_{R}^{*}}

is an isomorphism.

For

ℓ \neq p

, the modern developments of higher category theory allow one to define the

\infty

-category of Ã©tale

F_{ℓ}

-sheaves on any prestack (e.g., see [35]). In particular, for

Λ = F_{ℓ}, Z_{ℓ}, Q_{ℓ}

(or finite extension of these rings), it is possible to define the

\infty

1.2. Tamely ramified local geometric Langlands correspondence

We first recall the classical theory. Assume that

F

is a finite extension of

Q_{p}

or is isomorphic to

F_{q} ((ϖ))

, and for simplicity assume that

G

extends to a connected reductive group over

O

. (In fact, results in the subsection hold in appropriate forms for quasi-split groups that are split over a tamely ramified extension of

F

.) In addition, we fix a pinning

(B, T, e)

G

over

O

The classical local Langlands program aims to classify (smooth) irreducible representations of

G (F)

(over

C

) in terms of Galois representations. From this point of view, the Satake isomorphism (1.3) gives a classification of irreducible unramified representations of

G (F)

, i.e., those that have nonzero vectors fixed by

G (O)

, as such representations are in one-to-one correspondence with simple modules of

H_{G (O)}^{c l} \otimes C

, which via the Satake isomorphism (1.3) are parameterized by semisimple

\hat{G}

-conjugacy classes in

^{c} G

. (For an irreducible unramified representation

π

, the corresponding

\hat{G}

-conjugacy class in

^{c} G

is usually called the Satake parameter of

π

The next important class of irreducible representations are those that have nonzero vectors fixed by an Iwahori subgroup

G (F)

. For example, under the reduction mod

ϖ

map

G (O) \to G (k)

, the preimage

I

B (k) \subset G (k)

is an Iwahori subgroup of

G (F)

. As in the unramified case, the

Z

-valued

I

-biinvariant functions form a

Z

-algebra

H_{I}^{c l}

with multiplication given by convolution (1.2) (with the Haar measure

d g

chosen so that the volume of

I

is one), and the set of irreducible representations of

G (F)

that have nonzero

I

-fixed vectors are in bijection with the set of simple

(H_{I}^{c l} \otimes C)

-modules. Just as the Satake isomorphism, Kazhdan-Lusztig gave a description of

H_{I}^{c l} \otimes C

in terms of geometric objects associated to

\hat{G}

Let

\hat{U} \subset \hat{B}

denote the unipotent radical of

\hat{B}

. The natural morphism

\hat{U} / \hat{B} \to \hat{G} / \hat{G}

is usually called the Springer resolution. Let

S_{\hat{G}}^{u n i p} = (\hat{U} / \hat{B}) \times_{\hat{G} / \hat{G}} (\hat{U} / \hat{B})

and therefore on

S_{\hat{G}, C}^{unip}

, by identifying

{\hat{U}}_{C}

with its Lie algebra via the exponential map. Then
one can form the quotient stack

S_{\hat{G}, C}^{unip} / G_{m, C}

. In the sequel, for an Artin stack

X

(of finite presentation) over

C

, we let

K (X)

denote the

K

-group of the

(\infty

-)category of coherent sheaves on

X

Kazhdan-Lusztig [36] constructed (under the assumption that

G

is split with connected center) a canonical isomorphism (after choosing a square root of

\sqrt{q}

q

)

\begin{matrix} (1.7) & K (S_{\hat{G}, C}^{unip} / G_{m, C}) \otimes_{K (B G_{m, C})} C ≅ H_{I}^{c l} \otimes C \end{matrix}

where the map

K (B G_{m, C}) \to C

sends the class corresponding to the tautological representation of

G_{m, C}

\sqrt{q}

. In addition, the isomorphism induces the Bernstein isomorphism

\begin{matrix} (1.8) & K (B {\hat{G}}_{C}) \otimes C ≅ Z (H_{I}^{c l} \otimes C) \end{matrix}

where

Z (H_{I}^{c l} \otimes C)

is the center of

H_{I}^{c l} \otimes C

, and the map

K (B {\hat{G}}_{C}) \to K (S_{\hat{G}, C}^{unip} / G_{m, C})

is induced by the natural projection

S_{\hat{G}}^{unip} / G_{m} \to B \hat{G}

Remark 1.2.1. It would be interesting to give a description of the

Z

-algebra

H_{I}^{c l}

in terms of the geometry involving

\hat{G}

, which would generalize the integral Satake isomorphism from [83].

It turns out that the Kazhdan-Lusztig isomorphism (1.7) also admits a categorification, usually known as the Bezrukavnikov equivalence, which gives two realizations of the affine Hecke category. Again, when switching to the geometric theory, we allow

F

to be a little bit more general as in Section 1.1. We also assume that

G

extends to a connected reductive group over

O

and fix a pinning of

G

over

O

. Let

L^{+} G \to G_{k}

be the natural "reduction mod

ϖ

" map, and let Iw

\subset L^{+} G

be the preimage of

B_{k} \subset G_{k}

. This is the geometric analogue of

I

. Then as in the unramified case discussed in Section 1.1, one can define the Iwahori local Hecke stack

{H k}_{I w} = I w ∖ L G / I w

and the monoidal categories

{Shv}_{c} ({H k}_{I w} \otimes \bar{k}, Λ) \subset S h v ({H k}_{I w} \otimes \bar{k}, Λ)

. The category

S h v v_{c} ({H k}_{I w} \otimes \bar{k}, Λ)

can be thought as a categorical analogue of

H_{I}^{c l}

, usually called the affine Hecke category.

Recall that we let

\overset{˘}{F} = W_{O} (\bar{k}) [1 / ϖ]

. The inertia

I_{F} := Γ_{\overset{˘}{F}}

F

has a tame quotient

I_{F}^{t}

isomorphic to

\prod_{ℓ \neq p} Z_{ℓ} (1)

Theorem 1.2.2. For every choice of a topological generator

τ

of the tame inertia

I_{F}^{t}

, there is a canonical equivalence of monoidal

\infty

-categories

{Bez}_{G}^{unip} : Coh (S_{\hat{G}, Q_{ℓ}}^{unip}) ≅ {Shv}_{c} ({H k}_{I w} \otimes \bar{k}, Q_{ℓ})

In fact, Bezrukavnikov proved such equivalence when

F = k ((ϖ))

in [9]. Yun and the author deduced the mixed characteristic case from the equal characteristic case. It would be interesting to know whether the new techniques introduced in [25,59] can lead a direct proof of this equivalence in mixed characteristic. (See [1] for some progress in this direction.)

Remark 1.2.3. Again, for arithmetic applications, one needs to describe the action of

Γ_{k}

S h v_{c} ({H k}_{I w} \otimes \bar{k}, Λ)

in terms of the dual group side. See

[9, 35]

for a discussion.

We explain an important ingredient in the proof of Theorem 1.2.2 (when

F = k ((ϖ)))

. There is a smooth affine group scheme

E

(called the Iwahori group scheme)

over

D

, analogous to

{H k}_{G, D^{S}}

as discussed at the end of Section 1.1 (here

S = {1}

). In addition,

{{{H k}_{G, D} |}_{D^{*}} ≅ {H k}_{G, D} |}_{D^{*}}

and

{{H k}_{G, D} |}_{0} = {H k}_{I w}

, where

0 \in D

is the closed point. Then taking nearby cycles gives

\begin{matrix} (1.9) & Z : Coh {(B {\hat{G}}_{Λ})}^{◯} \overset{{Sat}_{11}}}{\to} Perv ({{H k}_{E, D} |}_{D_{k}^{*}}, Λ) \overset{Ψ}{\to} Perv ({H k}_{I w} \otimes \bar{k}, Λ) \end{matrix}

This is known as Gaitsgory's central functor [27,75], which can be regarded as a categorification of (1.8). We remark this construction is motivated by the Kottwitz conjecture originated from the study of mod

p

geometry of Shimura varieties. See Section 3.1 for some discussions.

Theorem 1.2.2 admits a generalization to the tame level. We consider the following diagram:

\hat{G} / \hat{G} \leftarrow \hat{B} / \hat{B} \overset{q_{\hat{B}}}{\to} \hat{T} / \hat{T}

where the left morphism is the usual Grothendieck-Springer resolution. Let

χ

be a

Λ

-point of

\hat{T} / \hat{T}

, where

Λ

is a finite extension of

Q_{ℓ}

. Let

(\hat{B} / \hat{B})_{χ} = q_{\hat{B}}^{- 1} (χ)

, and let

S_{\hat{G}, Λ}^{χ} := (\hat{B} / \hat{B})_{χ} \times_{\hat{G} / \hat{G}} (\hat{B} / \hat{B})_{χ}

Note that if

χ = 1

, this reduces to

S_{\hat{G}, Λ}^{unip}

. On the other hand, a (torsion)

Λ

-point

χ \in \hat{T} / \hat{T}

defines a one-dimensional character sheaf

L_{χ}

on Iw

\otimes \bar{k}

. Then one can define the monoidal category of bi-(Iw,

L_{χ}

)-equivariant constructible sheaves on

L G_{\bar{k}}

, denoted as

{Sh}_{cons} (χ {({H k}_{I w})}_{χ}, Λ)

. If

χ = 1

, so

L_{χ}

is the trivial character sheaf on Iw, this reduces to the affine Hecke category

{Shv}_{c} ({H k}_{I w} \otimes \bar{k}, Λ)

. The following generalization of Theorem 1.2.2 is conjectured in [9] and will be proved in a forthcoming joint work with Dhillon-Li-Yun [18].

Theorem 1.2.4. Assume that char

F = char k

. There is a canonical monoidal equivalence

{Bez}_{\hat{G}}^{χ} : Coh ({\hat{S}}_{\hat{G}, Λ}^{χ}) ≅ {Shv}_{c} ({({H k}_{I w})}_{χ}, Λ)

Remark 1.2.5. It is important to establish a version of equivalences in Theorems 1.2.2 and 1.2.4 for

Z_{ℓ}

-sheaves.

Remark 1.2.6. The local geometric Langlands correspondence beyond the tame ramification has not been fully understood, although certain wild ramifications have appeared in concrete problems (e.g., [31,79]). It is widely believed that the general local geometric Langlands should be formulated as 2-categorical statement, predicting the 2-category of module categories under the action of (appropriately defined) category of sheaves on

L G

is equivalent to the 2-category of categories over the stack of local geometric Langlands parameters. The precise formulation is beyond the scope of this survey, but, roughly speaking, it implies (and is more or less equivalent to saying) that the Hecke category for appropriately chosen "level" of

L G

is (Morita) equivalent to the category of coherent sheaves on some stack of the form

X \times_{Y} X

, where

Y

is closely related to the moduli of local geometric Langlands parameters.

1.3. Global geometric Langlands correspondence

As mentioned at the beginning of the article, the (global) geometric Langlands program originated from Drinfeld's proof of Langlands conjecture for

{G L}_{2}

over function fields. Early developments of this subject mostly focused on the construction of Hecke eigensheaves associated to Galois representations of a global function field

F

(or, equivalently, local systems on a smooth algebraic curve

X

), e.g., see

[20, 26, 42]

The scope of the whole program then shifted after the work [5], in which BeilinsonDrinfeld formulated a rough categorical form of the global geometric Langlands correspondence. The formulation then was made precise by Arinkin-Gaitsgory in [2], which we now recall. Let

X

be a smooth projective curve over

F = C

. On the automorphic side, let

D_{c} ({B u n}_{G})

be the

\infty

-category of coherent D-modules on the moduli stack Bun

_{G}

of principal

G

-bundles on

X

. On the Galois side, let

Coh ({Loc}_{\hat{G}})

be the

\infty

-category of coherent sheaves on the moduli stack

{Loc}_{\hat{G}}

of de Rham

\hat{G}

-local systems (also known as principal

\hat{G}

-bundles with flat connection) on

X

Conjecture 1.3.1. There is a canonical equivalence of $\infty$ -categories

L_{G} : Coh ({Loc}_{\hat{G}}) ≅ D_{c} ({Bun}_{G})

satisfying a list of natural compatibility conditions.

We briefly mention the most important compatibility condition, and refer to [2] for the rest. Note that both sides admit actions by a family of commuting operators labeled by

x \in X

and

V \in Coh {(B {\hat{G}}_{C})}^{\infty}

. Namely, for every point

x \in X

, there is the evaluation map

{Loc}_{\hat{G}} \to B {\hat{G}}_{C}

so every

V \in Coh {(B {\hat{G}}_{C})}^{⊙}

gives a vector bundle

{\tilde{V}}_{x}

{Loc}_{\hat{G}}

by pullback, which then acts on

Coh ({Loc}_{\hat{G}})

by tensoring. On the other hand, there is the Hecke operator

H_{V, x}

that acts on

D_{c} ({Bun}_{G})

by convolving the sheaf

{{Sat}_{{1}} (V) |}_{x}

from the (

D

-module version of) the geometric Satake (1.6). Then the equivalence

L_{G}

should intertwine the actions of these operators.

Although the conjecture remains widely open, it is known that the category of perfect complexes

Perf ({Loc}_{\hat{G}})

{Loc}_{\hat{G}}

acts on

D_{c} ({Bun}_{G})

, usually called the spectral action, in a way such that the action of

{\tilde{V}}_{x} \in Perf ({Loc}_{\hat{G}})

D_{c} ({Bun}_{G})

is given by the Hecke operator

H_{V, x}

Nowadays, Conjecture 1.3.1 sometimes is referred as the de Rham version of the global geometric Langlands conjecture, as there are some other versions of such conjectural equivalences, which we briefly mention.

First, in spirit of the nonabelian Hodge theory, there should exist a classical limit of Conjecture 1.3.1, sometimes known as the Dolbeault version of the global geometric Langlands. While the precise formulation is unknown (to the author), generically, it amounts to the duality of Hitchin fibrations for

G

and

\hat{G}

(in the sense of mirror symmetry), and was established "generically" in [15,19]. By twisting/deforming such duality in positive characteristic, one can prove a characteristic

p

analogue of Conjecture 1.3.1 (of course, only "generically," see

[10, 14, 15])

. Interestingly, this "generic" characteristic

p

equivalence can be used to give a new proof of the main result of [5] (at least for

G = {G L}_{n}

, see [12]).

The work [5] (and therefore the de Rham version of the global geometric Langlands) was strongly influenced by conformal field theory. On the other hand, motivated by topological field theory, Ben-Zvi and Nadler [7] proposed a Betti version of Conjecture 1.3.1, where on the automorphic side the category of

D

-modules on

{B u n}_{G}

is replaced with the category of sheaves of

C

-vector spaces on (the analytification of) Bun

_{G}

and on the Galois side

{Loc}_{\hat{G}}

is replaced by the moduli of Betti

\hat{G}

-local systems (also known as

\hat{G}

-valued representations of fundamental group of

X

The Riemann-Hilbert correspondence allows passing between the de Rham

\hat{G}

-local systems and Betti

\hat{G}

-local systems, but in a transcendental way. So Conjecture 1.3.1 and its Betti analogue are not directly related. Recently, Arinkin et al. [3] proposed yet another version of Conjecture 1.3.1, which directly relates both de Rham and Betti versions, and at the same time includes a version in terms of

ℓ

-adic sheaves. So it is more closely related to the classical Langlands correspondence over function fields, as will be discussed in Section 2.2.

2. FROM GEOMETRIC TO CLASSICAL LANGLANDS PROGRAM

In the previous section, we discussed how the ideas of categorification and geometrization led to the developments of the geometric Langlands program. On the other hand, the ideas of quantum physics allow one to reverse arrows in (1.1) by evaluating a (topological) quantum field theory at manifolds of different dimensions. Such ideas are certainly not new in geometry and topology. But, surprisingly, they also lead to a new understanding of the classical Langlands program. Indeed, it has been widely known that there is an analogy between global fields and 3-manifolds, and under such analogy Frobenius corresponds to the fundamental group of a circle. Then "compactification of field theories on a circle" leads to the categorical trace method (e.g., see

[3, 6, 77]

), which plays a more and more important role in the geometric representation theory.

2.1. Categorical arithmetic local Langlands

In this subsection, let

F

be either a finite extension of

Q_{p}

or isomorphic to

F_{q} ((ϖ))

. The classical local Langlands correspondence seeks a classification of smooth irreducible representations of

G (F)

in terms of Galois data. The precise formulation, beyond the

G = {G L}_{n}

case (which is a theorem by

[30, 43]

), is complicated. However, the yoga that the local geometric Langlands is 2-categorical (see Remark 1.2.6) suggests that the even the classical local Langlands correspondence should and probably needs to be categorified.

The first ingredient needed to formulate the categorical arithmetic local Langlands is the following result, due independently to

[17, 25, 82]

. We take the formulation from [82] and refer for the notion of (strongly) continuous homomorphisms to the same reference.

Theorem 2.1.1. The prestack sending a

Z_{ℓ}

-algebra

A

to the space of (strongly) continuous homomorphisms

ρ : W_{F} \to^{c} G (A)

such that

d \circ ρ = ({c y c l}^{- 1}, p r)

is represented by a (classical) scheme

{Loc}_{c}^{◻}

, which is a disjoint union of affine schemes that are flat, of finite type, and of locally complete intersection over

Z_{ℓ}

The conjugation action of

\hat{G}

^{c} G

induces an action of

\hat{G}

{Loc}_{c}^{◻}

, and we call the quotient stack

{Loc}_{_{G}} = {Loc}_{_{c}}^{◻} / \hat{G}

the stack of local Langlands parameters, which, roughly speaking, classifies the groupoid of the above

ρ

's up to

\hat{G}

-conjugacy.

In the categorical version of the local Langlands correspondence, on the Galois side it is natural to consider the

(\infty

-)category

Coh ({Loc}_{c})

of coherent sheaves on

{Loc}_{c}_{G}

. On the representation side, one might naively consider the

(\infty

-)category

R e p (G (F), Λ)

of smooth representations of

G (F)

. But in fact, this category needs to be enlarged. This can be seen from different point of view. Indeed, it is a general wisdom shared by many people that in the classical local Langlands correspondence, it is better to study representations of

G

together with a collection of groups that are (refined version of) its inner forms. There are various proposals of such collections. Arithmetic geometry (i.e., the study of

p

-adic and mod

p

geometry of Shimura varieties and moduli of Shtukas) and geometric representation theory (i.e., the categorical trace construction) suggest studying a category glued from the categories of representations of a collection of groups

{J_{b} (F)}_{b \in B (G)}

arising from the Kottwitz set

B (G) = G (\overset{˘}{F}) / \sim, where g \sim g^{'} if g^{'} = h^{- 1} g σ (h) for some h \in G (\overset{˘}{F})

Here for

b \in B (G)

(lifted to an element in

G (\overset{˘}{F})

), the group

J_{b}

is an

F

-group defined by assigning and

F

-algebra the group

J_{b} (R) = {h \in G (\overset{˘}{F} \otimes_{F} R) ∣ h^{- 1} b σ (h) = b}

. In particular, if

b = 1

then

J_{b} = G

. In general, there is a well-defined embedding

{(J_{b})}_{\bar{F}} \to G_{\bar{F}}

up to conjugacy, making

J_{b}

a refinement of an inner form of a Levi subgroup of

G

(say,

G

is quasisplit).

There are two ways to make this idea precise. One is due to Fargues-Scholze [25], who regard

B (G)

as the set of points of the

v

-stack

{Bun}_{G}

G

-bundles on the FarguesFontaine curve and consider the category

D_{l i s} ({Bun}_{G}, Λ)

of appropriately defined Ã©tale sheaves on

{Bun}_{G}

, which indeed glues all

Rep (J_{b} (F), Λ)

's together. We mention that this approach relies on Scholze's work on

ℓ

-adic formalism of diamond and condensed mathematics.

In another approach

[35, 64, 77, 82]

, closely related to the idea of categorical trace, the set

B (G)

is regarded as the set of points of the (Ã©tale) quotient stack

B (G) := L G / {Ad}_{σ} L G

where

{Ad}_{σ}

denotes the Frobenius twisted conjugation given by

{Ad}_{σ} : L G \times L G \to L G, (h, g) \mapsto h g σ (h)^{- 1}

Then we have the category of

Λ

-sheaves

S h v (B (G) \otimes \bar{k}, Λ)

as mentioned before. Although

B (G)

is a wild object in the traditional algebraic geometry, there are still a few things one can say about its geometry, and the category

Shv (B (G) \otimes \bar{k}, Λ)

is quite reasonable. In addition, it is possible to define the category

{S h v}_{c} (B (G) \otimes \bar{k}, Λ)

of constructible sheaves on

B (G) \otimes \bar{k}

, as we now briefly explain and refer to [35] for careful discussions.

For every algebraically closed field

Ω

over

k

, the groupoid of

Ω

-points of

B (G)

is the groupoid of

F

-isocrystals with

G

-structure over

Ω

and the set of its isomorphism classes can be identified with the Kottwitz set

B (G)

. However,

B (G)

is not merely a disjoint union
of its points. Rather, it admits a stratification, known as the Newton stratification, labeled by

B (G)

. Namely, the set

B (G)

has a natural partial order and, roughly speaking, for each

b \in B (G)

those

Ω

-points corresponding to

b^{'} \leq b

form a closed substack

i_{\leq b} : B (G)_{\leq b} \subset

B (G) \otimes \bar{k}

and those

Ω

-points corresponding to

b

form an open substack

j_{b} : B (G)_{b} \subset

B (G)_{\leq b}

. In particular, basic elements in

B (G)

(i.e., minimal elements with respect to the partial order

\leq

) give closed strata. We also mention that if

b

is basic,

J_{b}

is a refinement of an inner form of

G

, usually called an extended pure inner form of

G

In the rest of this subsection, we simply denote

B (G) \otimes \bar{k}

B (G)

. We write

i_{b} = i_{\leq b} j_{b} : B (G)_{b} ↪ B (G)

for the locally closed embedding. For

b

, let

{Rep}_{f.g.} (J_{b} (F), Λ)

be the full subcategory of

Rep (J_{b} (F), Λ)

generated (under finite colimits and retracts) by compactly induced representations

δ_{K, Λ} := c - {ind}_{K}^{J_{b} (F)} (Λ)

from the trivial representation of open compact subgroups

K \subset J_{b} (F)

. The following theorem from [35] summarizes some properties of

{Shv}_{c} (B (G), Λ)

Theorem 2.1.2. (1) An object in

Shv (B (G), Λ)

is constructible if and only if its !-restriction to each

B (G)_{b}

is constructible and is zero for almost all b's. If

Λ

is a field of characteristic zero,

{Shv}_{c} (B (G), Λ)

consist of compact objects in

Shv (B (G), Λ)

(2) For every

b \in B (G)

, there is a canonical equivalence

{Shv}_{c} (B (G)_{b}, Λ) ≅

{Rep}_{f.g.} (J_{b} (F), Λ)

. There are fully faithful embeddings

i_{b, *}, i_{b,!}

{Shv}_{c} (B (G)_{b}, Λ) \to {Shv}_{c} (B (G), Λ)

(which coincide when

b

is basic), inducing a semiorthogonal decomposition of

{Shv}_{c} (B (G), Λ)

in terms of

{{S h v}_{c} (B (G)_{b}, Λ)}_{b}

(3) There is a self-duality functor

D^{coh} : {Shv}_{c} (B (G), Λ) ≃ {S h v}_{c} (B (G), Λ)^{\lor} o b -

tained by gluing cohomological dualities (in the sense of Bernstein-Zelevinsky) on various

{Rep}_{f.g.} (J_{b} (F), Λ)

's.

(4) There is a natural perverse

t

-structure obtained by gluing (shifted)

t

-structures on various

{Rep}_{f.g.} (J_{b} (F), Λ)

's, preserved by

D^{coh}

Λ

is a field.

The following categorical form of the arithmetic local Langlands conjecture [82, SEcT. 4.6] is inspired by the global geometric Langlands conjecture as discussed in Section 1.3 .

Conjecture 2.1.3. Assume that

G

is quasisplit over

F

equipped with a pinning

(B, T, e)

and fix a nontrivial additive character

ψ : F \to Z_{ℓ} {[μ_{p^{\infty}}]}^{\times}

. There is a canonical equivalence of categories

Remark 2.1.4. (1) There is a closely related version of the conjecture, with

{Shv}_{c} (B (G), Λ)

replaced by

Shv (B (G), Λ)

and with

Coh ({Loc}_{G} \otimes Λ

) replaced by its ind-completion (with
certain support condition imposed) (see [82, SECT. 4.6]). Fargues-Scholze [25] make a conjecture parallel to this version, with the category

Shv (B (G), Λ)

replaced by

D_{lis} ({Bun}_{G}, Λ)

as mentioned above.

(2) It is also explained in [82] a motivic hope to have a version of such equivalence over

Q

One consequence of the conjecture is that for every

b

there should exist a fully faithful embedding

U_{J_{b}} : {Rep}_{f.g.} (J_{b} (F), Λ) \to Coh ({Loc}_{c}_{G} \otimes Λ)

obtained as the restriction of a quasinverse of

L_{G}

i_{b,!} ({Rep}_{f . g .} (J_{b} (F), Λ))

. The existence of such functor is closely related to the idea of local Langlands in families and has also been considered (in the case

J_{b} = G

is split and

Λ

is a field of characteristic zero) in [6,32].

In particular, for every open compact subgroup

K \subset J_{b} (F)

there should exist a coherent sheaf

\begin{matrix} (2.1) & V_{K, Λ} := U_{J_{b}} (δ_{K, Λ}) \end{matrix}

{Loc}_{G} \otimes Λ

such that

The algebra

H_{K, Λ}

is sometimes called the derived Hecke algebra as it might not concentrate on cohomological degree zero (when

Λ = Z_{ℓ}

F_{ℓ}

). See [82, SECTS. 4.3-4.5] for conjectural descriptions of

A_{K, Λ}

in various cases.

As in the global geometric Langlands conjecture, the equivalence from Conjecture 2.1.3 should satisfy a set of compatibility conditions. For example, it should be compatible with parabolic inductions on both sides, and should be compatible with the duality

D^{coh}

{Shv}_{c} (B (G), Λ)

and the (modified) Grothendieck-Serre duality of

Coh ({Loc}_{c} \otimes Λ)

. We refer to

[35, 82]

for more details.

On the other hand, Conjecture 2.1.3 predicts an action of the category

Perf ({Loc}_{G} \otimes Λ)

of perfect complexes on

{Loc}_{G} \otimes Λ

{Shv}_{c} (B (G), Λ)

, analogous to the spectral action as mentioned in Section 1.3. One of the main results of [25] is the construction of such action in their setting. Currently the existence of such a spectral action on

{Shv}_{c} (B (G), Λ)

is not known. But there are convincing evidences that Conjecture 2.1.3 should still be true.

We assume that

G

extends to a reductive group over

O

as before. Then there are closed substacks

usually called the stack of unramified parameters (resp. unipotent parameters), classifying those

ρ

such that

ρ (I_{F})

is trivial (resp.

ρ (I_{F})

is unipotent). For

Λ = Q_{ℓ}, {Loc}_{c}^{unip} \otimes Q_{ℓ}

is a connected component of

{Loc}^{G}_{G} \otimes Q_{ℓ}

On the other hand, there is the unipotent subcategory

{Shv}_{c}^{unip} (B (G), Q_{ℓ}) \subset

{Sh}_{c} (B (G), Q_{ℓ})

, which roughly speaking is the glue of categories

{Rep}_{f.g.}^{unip} (J_{b} (F), Q_{ℓ})

of unipotent representations of

J_{b} (F)

(introduced in [52]) for all

b \in B (G)

. We have the following theorem from [35], deduced from Theorem 1.2.2 by taking the Frobenius-twisted categorical trace.

Theorem 2.1.5. For a reductive group

G

over

O

with a fixed pinning

(B, T, e)

, there is a canonical equivalence

L_{G}^{unip} : Coh ({Loc}_{c}^{unip} \otimes Q_{ℓ}) ≅ {Sh}_{c}^{unip} (B (G), Q_{ℓ})

For arithmetic applications, it is important to match specific objects under the equivalence. We give a few examples and refer to [35] for many more of such matchings (see also [82, SECTS. 4.3-4.5]).

Example 2.1.6. The equivalence

L_{G}^{unip}

gives the the conjectural coherent sheaf in (2.1) for all parahoric subgroups

K \subset G (F)

(in the sense of Bruhat-Tits) such that (2.2) holds. For example, we have

V_{G (O), Q_{ℓ}} ≅ O_{{Loc}_{C_{G}}^{u r}} \otimes Q_{ℓ}

, which gives

\begin{matrix} (2.3) & (R {End}_{Coh ({Loc}_{G})} O_{{Loc}_{C G}^{u r}}^{)^{o p}} \otimes Q_{ℓ} ≅ {(R End δ_{G (O), Q_{ℓ}})}^{o p} = H_{G (O), Q_{ℓ}} ≅ H_{G (O)}^{c l} \otimes Q_{ℓ} \end{matrix}

{Loc}_{c G}^{u r} ≅ ({^{c} G |}_{d = (q, σ)}) / \hat{G}

, taking the 0 th cohomology recovers the Satake isomorphism (1.3). In addition, it implies that the left-hand side has no higher cohomology, which is not obvious. We mention that it is conjectured in [82, SECT. 4.3] that

V_{G (O), Z_{ℓ}} ≅ O_{{Loc}_{c_{G}}}

so the first isomorphism in (2.3) should hold over

Z_{ℓ}

, known as the (conjectural) derived Satake isomorphism. (But

H_{G (O), Z_{ℓ}} \neq H_{G (O)}^{c l} \otimes Z_{ℓ}

in general.)

There is also a pure Galois side description of

A_{I, Q_{ℓ}}

, known as the unipotent coherent Springer sheaf as defined in [6,82] (see also [32]).

Example 2.1.7. By construction, there is a natural morphism of stacks

{Loc}_{G} \to B \hat{G}

over

Z_{ℓ}

. For a representation of

\hat{G}

on a finite projective

Λ

-module, regarded as a vector bundle on

B {\hat{G}}_{Λ}

, let

\tilde{V}

be its pullback to

{Loc}_{G} \otimes Λ

, and let

\tilde{V}^{?} \in Perf ({Loc}_{c_{G}} \otimes Λ)

be its

We have

L_{G}^{u n i p} ({\tilde{V}}^{u r}) ≅ {N t}_{!} r! Sat (V) =: Φ_{V}

where

r

and

N t

are maps in the following correspondence:

{H k}_{G} = L^{+} G ∖ L G / L^{+} G \overset{r}{\leftarrow} L G / {Ad}_{σ} L^{+} G \overset{N t}{\to} L G / {Ad}_{σ} L G = B (G) .

In particular, for two representations

V

and

W

\hat{G}

, there is a morphism

\begin{matrix} (2.4) & R {Hom}_{{L o c}_{C G}^{u r}} \otimes Q_{ℓ} ({\tilde{V}}^{u r}, {\tilde{W}}^{u r}) \to R {Hom}_{{Shv}_{c} (B (G), Q_{ℓ}}) (S_{V}, ς_{W}) \end{matrix}

compatible with compositions. Such map was first constructed in [64,77] and (the version for underived Hom spaces) was then extended to

Z_{ℓ}

-coefficient in [70]. It has significant arithmetic applications, as will be explained in Section 3.

Remark 2.1.8. It is likely that Theorem 2.1 .5 can be extended to the tame level by taking the Frobenius-twisted categorical trace of the equivalence from Theorem 1.2.4. On the other hand, as mentioned in Remark 1.2.5, it is important to extend these equivalences to

Z_{ℓ^{-}}

coefficient.

2.2. Global arithmetic Langlands for function fields

Next we turn to global aspects of the arithmetic Langlands correspondence. As mentioned at the beginning, its classical formulation, very roughly speaking, predicts a natural correspondence between the set of (irreducible) Galois representations and the set of (cuspidal) automorphic representations. As in the local case, beyond the

{G L}_{n}

case (which is a theorem by [38]), such a formulation is not easy to be made precise. On the other hand, the global geometric Langlands conjecture from Section 1.3 and philosophy of decategorification/trace suggest that the global arithmetic Langlands can and probably should be formulated as an isomorphism between two vector spaces, arising from the Galois and the automorphic side, respectively. In this subsection, we formulate such a conjecture in the global function field case.

Let

F = F_{q} (X)

be the function field of a geometrically connected smooth projective curve

X

over

F_{q}

. We write

η = Spec F

for the generic point of

X

and

\bar{η}

for a geometric point over

η

. Let

| X |

denote the set of closed points of

X

. For

v \in | X |

, let

O_{v}

denote the complete local ring of

X

v

and

F_{v}

its fractional field. Let

O_{F} = \prod_{v \in | X |} O_{v}

be the integral adÃ¨les, and

A_{F} = \prod_{v \in | X |}^{'} F_{v}

the ring of adÃ¨les. For a finite nonempty set of places

Q

, let

W_{F, Q}

denote the Weil group of

F

, unramified outside

Q

Let

G

be a connected reductive group over

F

. Similarly to the local situation, the first step to formulate our global conjecture is the following theorem from [82].

Theorem 2.2.1. Assume that

ℓ ∤ 2 p

. The prestack sending

a Z_{ℓ}

-algebra

A

to the space of (strongly) continuous homomorphisms

ρ : W_{F, Q} \to^{c} G (A)

such that

d \circ ρ = ({c y c l}^{- 1}, p r)

is represented by a derived scheme

{Loc}_{c}^{◻}

, , which is a disjoint union of derived affine schemes that are flat and of finite type over

Z_{ℓ}

. If

Q \neq \emptyset, {Loc}_{c G, Q}^{◻}

is quasismooth.

We then define the stack of global Langlands parameters as

{Loc}_{G, Q} = {Loc}_{_{G}, Q} / \hat{G}

. Similar to the local case (see Example 2.1.7), for a representation of

{\hat{G}}_{Λ}

on a finite projective

Λ

-module, regarded as a vector bundle on

B {\hat{G}}_{Λ}

, let

\tilde{V}

be its pullback to

{Loc}_{G, Q} \otimes Λ

. If

V

is the restriction of a representation of

{(^{c} G)}^{S}

along the diagonal embedding

\hat{G} \to {(^{c} G)}^{S}

, then there is a natural (strongly) continuous

W_{F, Q}^{S}

-action on

\tilde{V}

(see [82, SEcT. 2.4]). For a place

v

F

, let

{Loc}_{_{c} G, v}

denote the stack of local Langlands parameters for

G_{F_{v}}

. Let

res : {Loc}_{G, Q} \to \prod_{v \in Q} {Loc}_{G, v}

denote the map by restricting global parameters to local parameters (induced by the map

W_{F_{v}} \to W_{F, Q})

. Later on, we will consider the !-pullback of coherent sheaves on

\prod_{v \in Q} {Loc}_{c, v}

along this map.

Remark 2.2.2. (1) In fact, when

Q = \emptyset

, the definition of

{Loc}_{G, Q}

needs to be slightly modified.

(2) Unlike the local situation,

{Loc}_{G}, Q

has nontrivial derived structure in general (see [82, REMARK 3.4.5]). Let

^{c l} {Loc}^{G, Q}^{Q}

denote the underlying classical stack.

(3) A different definition of

{Loc}_{G, Q} \otimes Q_{ℓ}

is given by [3].

Next we move to the automorphic side. For simplicity, we assume that

G

is split over

F_{q}

in this subsection. Fix a level, i.e., an open compact subgroup

K \subset G (O_{F})

. Let

Q

be the set of places consisting of those

v

such that

K_{v} \neq G (O_{v})

. For a finite set

S

, let Sht

_{K} (G)_{(X - Q)^{S}}

denote the ind-Deligne-Mumford stack over

(X - Q)^{S}

of the moduli of

G

-shtukas on

X

with

S

-legs in

X - Q

and

K

-level structure. (For example, see [39] for basic constructions and properties of this moduli space.) Its base change along the diagonal map

\bar{η} \to (X - Q) \overset{Δ}{\to} (X - Q)^{S}

is denoted by

{Sht}_{K} (G)_{Δ (\bar{η})}

. For every representation

V

{(^{c} G)}^{S}

on a finite projective

Λ

-module, the geometric Satake (1.6) (with

D

replaced by

X - Q

and with

Λ = Z_{ℓ}

allowed) provides a perverse sheaf

{Sat}_{S} (V)

{Sht}_{K} (G)_{(X - Q)^{S}}

. Let

C_{c} ({Sht}_{K} (G)_{Δ (\bar{η})}

, Sat

S_{S} (V))

denote the (cochain complex of the) total compactly supported cohomology of

{Sht}_{K} (G)_{Δ (\bar{η})}

with coefficient in

{Sat}_{S} (V)

. It admits a (strongly) continuous action of

W_{F, Q}^{S}

(see [34] for the construction of such action at the derived level, based on

[67, 68]

), as well as an action of the corresponding global (derived) Hecke algebra (with coefficients in

Λ

)

\begin{matrix} (2.5) & H_{K, Λ} = {(R End (c - {ind}_{K}^{G (A_{F})} (Λ)))}^{o p} \end{matrix}

For example, if

V = 1

is the trivial representation, then (under our assumption that

G

is split)

C_{c} ({Sht}_{K} (G)_{Δ (\bar{η})}, {Sat}_{{1}} (1)) = C_{c} (G (F) ∖ G (A) / K, Λ)

Here

G (F) ∖ G (A) / K

is regarded as a discrete DM stack over

\bar{η}

, and

C_{c} (G (F) ∖ G (A) / K, Λ)

denotes its compactly supported cohomology. When

Λ = Q_{ℓ}

, this is the space of compactly supported functions on

G (F) ∖ G (A) / K

We will fix a pinning

(B, T, e)

G

and a nondegenerate character

ψ : F ∖ A \to

Z_{ℓ} {[μ_{p}]}^{\times}

, which gives the conjectural equivalence

L_{v}

as in Conjecture 2.1.3 for every

v \in Q

. In particular, corresponding to

K_{v} \subset G (F_{v})

there is a conjectural coherent sheaf

N_{K_{v}}

(see (2.1)) on

{Loc}_{G, v}

Conjecture 2.2.3. There is a natural

(W_{F, Q}^{S} \times H_{K, Λ})

-equivariant isomorphism

We refer to [82, SECT. 4.7] for more general form of the conjecture (where "generalized level structures" are allowed) and examples of such conjecture in various special cases. This conjecture could be regarded a precise form of the global Langlands correspondence for function fields. Namely, it gives a precise recipe to match Galois representations and automorphic representations. (For example, V. Lafforgue's excursion operators are encoded in such isomorphism, see below.) Moreover, such an isomorphism fits in the Arthur-Kottwitz multiplicity formula and at the same time extends such a formula to the integral level and therefore relates to automorphic lifting theories.

The most appealing evidence of this conjecture is the following theorem [40,82], as suggested (at the heuristic level) by Drinfeld as an interpretation of Lafforgue's construction.

with an action of

H_{K, Q_{ℓ}}

, such that for every finite dimensional

Q_{ℓ}

-representation

V

{(^{c} G)}^{S}

, there is a natural

(W_{F, Q}^{S} \times H_{K, Q_{ℓ}})

-equivariant isomorphism

We mention that this theorem actually was proved for any

G

in [40,82]. In addition, when

K

is everywhere hyperspecial, (2.2.4) holds at the derived level by [3].

The isomorphism (2.2.4) induces an action of

Γ (^{c l} {Loc}^{c}_{G, Q} \otimes Q_{ℓ}, O)

on the righthand side. This is exactly the action by

V

. Lafforgue's excursion operators, which induces the decomposition of the right-hand side (in particular,

C_{c} (G (F) ∖ G (A_{F}) / K, Q_{ℓ})

) in terms of semisimple Langlands parameters. As explained [40], over an elliptic Langlands parameter, such an isomorphism is closely related to the Arthur-Kottwitz multiplicity formula. In the case of

G = {G L}_{n}

, it gives the following corollary, generalizing [38].

Corollary 2.2.5. Let

π

be a cuspidal automorphic representation of

{G L}_{n}

, with the associated irreducible Galois representation

ρ_{π} : W_{F, Q} \to {GL}_{n} (Λ)

for some finite extension

Λ / Q_{ℓ}

and with

m_{π}

the corresponding maximal ideal of

Γ (^{c l} {Loc}^{G}, Q \otimes Λ, O)

. Then there is an

(W_{F, Q}^{S} \times H_{K})

-equivariant isomorphism

H_{c}^{*} ({Sht}_{K} (G)_{Δ (\bar{η})}, {Sat}_{S} (V)) / m_{π} ≅ V_{ρ_{π}} \otimes π^{K}

In particular, the left-hand side only concentrates in cohomological degree zero.

2.3. Geometric realization of Jacquet-Langlands transfer

The global Langlands correspondence for number fields is far more complicated In fact, there are analytic part of the theory which currently seems not to fit the categorification/decategorification framework. Even if we just restrict to the algebraic/arithmetic part of the theory, there are complications coming from the place at

ℓ

and at

\infty

. In particular, the categorical forms of the local Langlands correspondence at

ℓ

and

\infty

are not yet fully understood.

Nevertheless, in a forthcoming joint work with Emerton and Emerton-Gee [21, 22], we will formulate conjectural Galois theoretical descriptions for the cohomology of Shimura varieties and even cohomology for general locally symmetric space, parallel to Conjecture 2.2.3. In this subsection, we just review a conjecture from [82] on the geometric realization of Jacquet-Langlands transfer via cohomology of Shimura varieties and discuss results from

[35, 64]

towards this conjecture.

We fix a few notations and assumptions. We fix a prime

p

in this subsection.

Let

A_{f} = \prod_{q}^{'} Q_{q}

denote the ring of finite adÃ¨les of

Q

, and

A_{f}^{p} = \prod_{q \neq p}^{'} Q_{p}

. We write

\bar{η} = Spec \bar{Q}

, where

\bar{Q}

is the algebraic closure of

Q

C

. For a Shimura datum

(G, X)

, let

μ

be the (minuscule) dominant weight of

\hat{G}

(with respect to

(\hat{B}, \hat{T})

) determined by

(G, X)

in the usual way and let

V_{μ}

denote the minuscule representation of

\hat{G}

of highest weight

μ

. Let

E \subset \bar{Q} \subset C

be the reflex field of

(G, X)

and write

d_{μ} = \dim X

. For a level (i.e., an open compact subgroup)

K = K_{p} K^{p} \subset G (Q_{p}) G (A_{f}^{p})

, let

{Sh}_{K} (G)

be the corresponding

Shimura variety of level

K

(defined over the reflex field

E

), and let

{Sh}_{K} (G)_{\bar{η}}

denote its base change along

E \to \bar{Q}

. Let

v

be a place of

E

above

p

. By a specialization

s p : \bar{η} \to \bar{v}

, we mean a morphism from

\bar{η}

to the strict henselianization of

O_{E}

v

To avoid many complications from Galois cohomology (e.g., the difference between extended pure inner forms and inner forms) and also some complications from geometry (e.g., the relation between Shimura varieties and moduli of Shtukas), we assume that

G

is of adjoint type in the rest of this subsection, and refer to [64] for general

G

. See also [82] with less restrictions on

G

Definition 2.3.1. Let

G

be a connected reductive group over

Q

. A prime-to-

p

(resp. finitely) trivialized inner form of

G

is a

G

-torsor

β

over

Q

equipped with a trivialization

β

over

A_{f}^{p}

(resp. over

A_{f}

). Then

G^{'} := Aut (ξ)

is an inner form of

G

(so the dual group of

G

and

G^{'}

are canonically identified), equipped with an isomorphism

θ : G (A_{f}^{p}) ≅ G^{'} (A_{f}^{p})

(resp.

θ : G (A_{f}) ≅ G^{'} (A_{f}))

Now let

(G, X)

and

(G^{'}, X^{'})

be two Shimura data, with

G^{'}

a prime-to-

p

trivialized inner form of

G

. Via

θ

, one can transport

K^{p} \subset G (A_{f}^{p})

to an open compact subgroup

K^{' p} \subset G^{'} (A_{f}^{p})

. We identify the prime-to-

p

(derived) Hecke algebra

H_{K^{p}, Λ}

(defined in the same way as in (2.5)) with

H_{K^{' p}, Λ}

and simply write them as

H_{K^{p}, Λ}

. Let

K_{p}^{'} \subset G^{'} (Q_{p})

be an open compact subgroup and write

K^{'} = K_{p}^{'} K^{' p}

for the corresponding level.

We fix a quasisplit inner form

G_{Q_{p}}^{*}

G_{Q_{p}}

and

G_{Q_{p}}^{'}

equipped with a pinning

(B_{Q_{p}}^{*}, T_{Q_{p}}^{*}, e^{*})

, and realize

G_{Q_{p}}

J_{b}

and

G_{Q_{p}}^{'}

J_{b^{'}}

for

b, b \in B (G_{Q_{p}}^{*})

. Under our assumption that

G

and

G^{'}

are adjoint, such

b, b^{'}

exist and are unique. Then we have the conjectural coherent sheaf

U_{K_{p}, Λ}

and

H_{K_{p}^{'}, Λ}

as in (2.1) on the stack Loc

_{G, p} \otimes Λ

of local Langlands parameters for

G_{Q_{p}}^{*}

over

Λ

Conjecture 2.3.2. For every choice of specialization map

s p : \bar{η} \to \bar{v}

, there is a natural map

\begin{aligned} R {Hom}_{Coh ({Locc}_{G, p} \otimes Λ)} (\tilde{V_{μ}} \otimes N_{K_{p}, Λ}, \tilde{V_{μ^{'}}} \otimes N_{K_{p}^{'}, Λ}) \\ (2.6) & \to R {Hom}_{H_{K} p, Λ} (C_{c} ({Sh}_{K} (G)_{\bar{η}}, Λ [d_{μ}]), C_{c} ({Sh}_{K^{'}} {(G^{'})}_{\bar{η}}, Λ [d_{μ^{'}}])) \end{aligned}

compatible with compositions. In particular, there is an (

E_{1^{-}}

)algebra homomorphism

compatible with (2.6). In addition, the induced action

\begin{matrix} (2.8) & H_{K_{p}, Λ} \overset{(2.2)}{≅} R End (U_{K_{p}, Λ}) \to R End (\tilde{V_{μ}} \otimes U_{K_{p}, Λ}) \overset{S}{\to} R {End}_{H_{K^{p}, Λ}} (C_{c} ({Sh}_{K} (G)_{\bar{η}}, Λ)) \end{matrix}

coincides with the natural Hecke action of

H_{K_{p}, Λ}

C_{c} ({Sh}_{K} (G)_{\bar{η}}, Λ)

(and therefore is independent of the specialization map

s p

This conjecture would be a consequence of a Galois theoretic description of

C_{c} ({Sh}_{K} (G)_{\bar{η}}, Λ)

similar to Conjecture 2.2.3, but its formulation does not require the existence of the stack of global Langlands parameters for

Q

. In any case, a step towards a Galois-theoretical description of

C_{c} ({Sh}_{K} (G)_{\bar{η}}, Λ)

might require Conjecture 2.3.2 as an
input. We also remark that as in the function field case, there is a more general version of such conjecture in [82, SECT. 4.7], allowing "generalized level structures," so that the cohomology of Igusa varieties could appear.

The following theorem verifies the conjecture in special cases.

Theorem 2.3.3. Suppose that the Shimura data

(G, X)

and

(G^{'}, X^{'})

are of abelian type, with

G^{'}

a finitely trivialized inner form of

G

. Suppose that

G_{Q_{p}}

is unramified (and therefore so is

G_{Q_{p}}^{'}

(1) The map (2.6) (and therefore (2.7)) exists when

Λ = Q_{ℓ}

and

K_{p} \subset G (Q_{p})

and

K_{p}^{'} \subset G^{'} (Q_{p})

are parahoric subgroups (in the sense of Bruhat-Tits).

(2) If

K_{p}

is hyperspecial, then the map (2.6) (and therefore (2.7)) exists when

Λ = Z_{ℓ}

, at least for underived Hom spaces. In addition, the action of

H_{K_{p}}^{c l}

H_{c}^{*} ({Sh}_{K} (G)_{\bar{η}}, Λ)

via (2.8) coincides with the natural action of

H_{K_{p}}^{c l}

Part (1) is proved in [35,64]. The proof contains two ingredients. One is the construction of physical correspondences between mod

p

fibers of

{S h}_{K} (G)

and

{S h}_{K^{'}} (G^{'})

by [64] (this is where we currently need to assume that

G

and

G^{'}

are unramified at

p

). The other ingredient is Theorem 2.1.5 (and therefore requires

Λ = Q_{ℓ}

). When

K_{p}

is hypersepcial, one can work with

Z_{ℓ}

-coefficient, as (the underived version of) (2.4) exists for

Z_{ℓ}

-coefficient thanks to [70]. In fact, in this case one can allow nontrivial local systems on the Shimura varieties (see [70]). The last statement is known as the

S = T

for Shimura varieties. The case when

d_{μ} = \dim {Sh}_{K} (G) = 0

is contained in [64]. The general case is proved in [63,74] using foundational works from

[25, 59]

3. APPLICATIONS TO ARITHMETIC GEOMETRY

Besides the previously mentioned directly applications of (ideas from) geometric Langlands to the classical Langlands program, we discuss some further arithmetic applications, mostly related to Shimura varieties and based on the author's works. We shall mention that there are many other remarkable applications of (ideas of) geometric Langlands to arithmetic problems, such as

[28, 31, 44, 66, 71]

, to name a few.

3.1. Local models of Shimura varieties

The theory of integral models of Shimura varieties (with parahoric level) started (implicitly in the work of Kronecker) with understanding of the

mod p

reduction of elliptic modular curves with

Γ_{0} (p)

-level. We discuss a small fraction of this theory concerning Ã©tale local structures of these integral models via the theory of local models. The recent developments of the theory of local models are greatly influenced by the geometric Langlands program.

We use notations from Section 2.3 for Shimura varieties (but we do not assume that

G

is of adjoint type in this subsection). Let

(G, X)

be a Shimura datum and

K

a chosen level with

K_{p} = E (Z_{p})

for some parahoric group scheme

E

(in the sense of Bruhat-Tits) of

G_{Q_{p}}

over

Z_{p}

. Then for a place

v

E

over

p

, a local model diagram is a correspondence of quasiprojective schemes over

O_{E_{v}}

\begin{matrix} (3.1) & S_{K} \leftarrow {\tilde{S}}_{K} \overset{\tilde{φ}}{\to} M_{G}^{l o c} \end{matrix}

where

S_{K}

is an integral model of

{Sh}_{K} (G)

over

O_{E_{v}}, {\tilde{S}}_{K}

is a

E_{O_{E_{v}}}

-torsor over

S_{K}, M_{G}^{loc}

is the so-called local model, which is a flat projective scheme over

O_{E_{v}}

equipped with a

E_{O_{E_{v}}}

-action, and

\tilde{φ}

is a

E_{O_{E_{v}}}

-equivariant smooth morphism of relative dimension dim

G

Therefore,

M_{G}^{loc}

models Ã©tale local structure of

S_{K}

. On the other hand, the existence of

U_{O_{E_{v}}}

-action on

M_{G}^{loc}

makes it easier than

S_{K}

to study.

The original construction of local models is based on realization of a parahoric group scheme as (the neutral connected component of) the stabilizer group of a self-dual lattice chain in a vector space (over a division algebra over

F

) with a bilinear form, e.g., see [57] for a survey and references. This approach is somehow ad hoc and is limited the so-called (P)EL (local) Shimura data. A new approach, based on the construction of an

Z_{p}

-analogue of the stack

{H k}_{g, D}

from Section 1.2, was systematically introduced in [58] (under the tameness assumption of

G

which was later lifted in [46,50]). In [58] the construction of such a

Z_{p}

-analogue (or rather the corresponding Beilinson-Drinfeld-type affine Grassmannian

Gr Z_{p}

over

Z_{p}

) is based on the construction of certain "two dimensional parahoric" group scheme

\tilde{E}

over

Z_{p} [ϖ]

whose restriction along

Z_{p} [ϖ] \overset{ϖ \mapsto p}{\to} Z_{p}

recovers

E

. (See [81] for a survey.) A more direct construction of a different

p

-adic version of such affine Grassman-

model is defined as the flat closure of the Schubert variety in the generic fiber corresponding to

μ

. In addition, the recent work [1] shows that the two constructions agree. The following theorem from [1] is the most up-to-date result on the existence of local models and about their properties.

Theorem 3.1.1. Let

G

be a connected reductive group over a p-adic field

F

. Except the odd unitary case when

p = 2

and triality case when

p = 3

, for every parahoric group scheme

E

G

over

O

, and a conjugacy class of minuscule cocharacters

μ

G

defined over a finite extension

E / F

F

, there is a normal flat projective scheme

M_{G, μ}^{l o c}

over

O_{E}

, equipped with a

E_{O_{E}}

-action such that

M_{Q, μ}^{l o c} \otimes E

G_{E}

-equivariantly isomorphic to the partial flag variety

F ℓ_{μ}

G_{E}

corresponding to

μ

, and that

M_{G}^{l o c} \otimes k_{E}

(G \otimes k_{E})

-equivariantly isomorphic to the (canonical deperfection of the) union over the

μ

-admissible set of Schubert varieties in

L G / L^{+} E \otimes k_{E}

. In addition,

M_{G}^{loc}

is normal, Cohen-Macaulay and each of its geometric irreducible components in its special fiber is normal and Cohen-Macaulay.

We end this subsection with a few remarks.

Remark 3.1.2. (1) Once the local model diagram (3.1) is established, this theorem also gives the corresponding properties of the integral models of Shimura varieties.

(2) A key ingredient in the study of special fibers of local models is the coherence conjecture by Pappas-Rapoport [56], proved in [75] (and the proof uses the idea of fusion).

(3) One important motivation/application of the theory of local models is the Haines-Kottwitz conjecture [29], which predicts certain central element in the parahoric Hecke algebra

H_{K_{p}}^{c l}

should be used as the test function in the trace formula computing the Hasse-Weil zeta function of

{S h}_{K} (G)

. As mentioned in Section 1.2, this conjecture motivated Gaitsgory's central sheaf construction (1.9). With the local Hecke stack

H k Z_{,} Z_{p}

over

Z_{p}

constructed (either the version from [58] or from [59]), one can mimic the construction (1.9) in mixed characteristic to solve the Kottwitz conjecture. Again, see [1] for the up-to-date result.

3.2. The congruence relation

We use notations and (for simplicity) keep assumptions from Section 2.3 regarding Shimura varieties. Let

(G, X)

be a Shimura datum abelian type, and let

K

be a level such that

K_{p}

is hyperspecial. Let

v ∣ p

be the place of

E

. Then

{Sh}_{K} (G)

has a canonical integral model

S_{K}

defined over

O_{E, (v)}

[37]. Let

{\bar{S}}_{K}

be its mod

p

fiber, which is a smooth variety defined over the residue field

k_{v}

v

. Let

σ_{v}

denote the geometric Frobenius in

Γ_{k_{v}}

. Theorem 2.3.3

consequences.

The congruence relation conjecture (also known as the Blasius-Rogawski conjecture), generalizing the classical Eichler-Shimura congruence relation Frob

_{p} = T_{p} + V_{p}

for modular curves, predicts that in the Chow group of

{\bar{S}}_{K} \times {\bar{S}}_{K}

, the Frobenius endomorphism of

{\bar{S}}_{K}

satisfies a polynomial whose coefficients are mod

p

reduction of certain Hecke correspondences. Theorem 2.3.3, together with [65, SECT. 6.3], implies this conjecture at the level of cohomology.

For every representation

V

^{c} (G_{Q_{p}})

, its character

χ_{V}

(regarded as a

\hat{G}

-invariant function on

{^{c} G |}_{d = (p, σ_{p})})

gives an element

h_{V} \in H_{G (Z_{p})}^{c l}

via the Satake isomorphism (1.3).

Theorem 3.2.1. The following identity,

\begin{matrix} (3.2) & \sum_{i = 0}^{n} (- 1)^{j} h_{χ_{\land} j_{V}} σ_{v}^{\dim V - j} = 0 \end{matrix}

Indeed, by [65, sEct. 6.3], such an equality holds with

h_{χ_{\land^{i} V}}

replaced by

S (χ_{\land^{i} V})

, where

S

is from Theorem 2.3.3 (1). Then part (2) of that theorem allows one to replace

S (χ_{\land^{i} V})

h_{χ_{\land^{i} V}}

. This approach to (3.2) is the Shimura variety analogue of V. Lafforgue's approach to the Eichler-Shimura relation for

{Sht}_{K} (G)

[39]. Traditionally, there is another approach to the congruence relation conjecture for Shimura varieties by directly studying reduction mod

p

of Hecke operators, starting from [24] for the Siegel modular variety case.

See [45] for the latest progress and related references. This approach would give (3.2) at the level of algebraic correspondences.

Now suppose

(G, X) = ({Res}_{F^{+} / Q} {(G_{0})}_{F^{+}}, \prod_{φ : F^{+} \to R} X_{0})

, where

(G_{0}, X_{0})

is a Shimura datum and

F^{+}

is a totally real field. As before, let

p

be a prime such that

K_{p}

is hyperspecial. In particular,

p

is unramified in

F^{+}

. In addition, for simplicity we assume that

G_{0, Q_{p}}

is split (so for a place

v

E

above

p, E_{v} = Q_{p}

). We let

F

denote an algebraic closure of

F_{p}

. Let

{w_{i}}_{i}

be the set of primes of

F^{+}

above

p

, and let

k_{i}

denote the residue field of

w_{i}

. For each

i

, we also fix an embedding

ρ_{i} : k_{i} \to F

. Then there is a natural map

where

S_{f_{i}}

is the permutation group on

f_{i}

letters. Together with Theorem 2.3.3, one obtains the following result [64].

Theorem 3.2.2. There is an action of

\prod_{i} (Z^{f_{i}} ⋊ S_{f_{i}})

H_{c}^{*} ({\bar{S}}_{K, \bar{F}}, Z_{ℓ})

such that action of

σ_{p}

factors as

σ_{p} = \prod_{i} σ_{p, i}

, where

σ_{p, i} = ((1, 0, \dots, 0), (12 \dots f_{i})) \in Z^{f_{i}} ⋊ S_{f_{i}}

. Each

σ_{p, i}^{f_{i}}

satisfies a polynomial equation similar to (3.2).

This theorem gives some shadow of the plectic cohomology conjecture of NekovÃ¡Å™Scholl [54].

3.3. Generic Tate cycles on mod $p$ fibers of Shimura varieties

In [64], we applied Theorem 2.3.3 to verify "generic" cases of Tate conjecture for the

mod p

fibers of many Shimura varieties. We use notations and (for simplicity) keep assumptions from Section 3.2. Let

{({\bar{S}}_{K, \bar{k_{v}}})}^{pf}

denote the perfection of

{\bar{S}}_{K, \bar{k_{v}}}

(i.e., regard it as a perfect presheaf over

A f f \frac{k f}{{\bar{k}}_{v}})

, then by attaching to every point of

{\bar{S}}_{K, \bar{k}}

F

-isocrystal with

G

-structure (see

[37, 64]

), one can define the so-called Newton map

N t : {({\bar{S}}_{K, \bar{k_{v}}})}^{p f} \to B {(G_{Q_{p}})}_{\bar{k_{v}}}

Then the Newton stratification of

B {(G_{Q_{p}})}_{\bar{k_{v}}}

(see Section 2.1) induces a stratification of

{\bar{S}}_{K, \bar{k_{v}}}

by locally closed subvarieties. It is known that the image of

N t

contains a unique basic element

b

and the corresponding subvarieties in

{\bar{S}}_{K, \bar{k_{v}}}

is closed, called the basic Newton stratum, and denoted by

{\bar{S}}_{b}

Let

m

be the order of the action of the geometric Frobenius

σ_{p}

X ∙ (\hat{T})

. Let

Λ_{p}^{T a t e} = {λ \in X^{∙} (\hat{T}) ∣ \sum_{i = 0}^{m - 1} σ_{p}^{i} (λ) = 0} \subset X^{∙} (\hat{T})

For a representation

V

{\hat{G}}_{Q_{ℓ}}

and

λ \in X ∙ (\hat{T})

, let

V (λ)

denote the

λ

-weight subspace of

V

(with respect to

\hat{T}

), and let

V^{Tate} = ⨁_{λ \in Λ_{p}^{Tate}} V (λ)

We are in particular interested in the condition

V_{μ}^{Tate} \neq 0

. As explained in the introduction of [64], under the conjectural Galois theoretic description of the cohomology of the

Shimura varieties (analogous to Conjecture 2.2.3), for a Hecke module

π_{f}

whose Satake parameter at

p

is general enough, certain multiple

a (π_{f})

of the dimension of this vector space should be equal to the dimension of the space of Tate classes in the

π_{f}

-component of the middle dimensional compactly-supported cohomology of

{\bar{S}}_{K, \bar{k_{v}}}

. In addition, this space is usually large. For example, in the case

G

is an odd (projective) unitary group of signature

(i, n - i)

over a quadratic imaginary field, the dimension of this space at an inert prime is

(\binom{\frac{n + 1}{2}}{i})

For a (not necessarily irreducible) algebraic variety

Z

of dimension

d

over an algebraically closed field, let

H_{2 d}^{B M} (Z) (- d)

denote the

(- d)

-Tate twist of the top degree BorelMoore homology, which is the vector space spanned by the irreducible components of

Z

. Now let

X

be a smooth variety of dimension

d + r

defined over a finite field

k

q

elements, and let

Z \subseteq X_{\bar{k}}

be a (not necessarily irreducible) projective subvariety of dimension

d

. There is the cycle class map

c l : H_{2 d}^{B M} (Z) (- d) \to ⋃_{j \geq 1} H_{c}^{2 d} {(X_{\bar{k}}, Q_{ℓ} (d))}^{σ_{q}^{j}} =: T_{ℓ}^{d} (X)

Theorem 3.3.1. We write

d_{μ} = \dim X = 2 d

and

r = \dim V_{μ}^{Tate}

(1) The basic Newton stratum

{\bar{S}}_{b}

{\bar{S}}_{K, {\bar{k}}_{v}}

is pure of dimension d. In particular,

d

is always an integer. In addition, there is an

H_{K, Q_{ℓ}}

-equivariant isomorphism

H_{2 d}^{B M} ({\bar{S}}_{b}) (- d) ≅ C {(G^{'} (Q) ∖ G^{'} (A_{f}) / K, Q_{ℓ})}^{\oplus r}

where

G^{'}

is the finitely trivialized inner form of

G

with

G_{R}^{'}

is compact.

(2) Let

π_{f}

be an irreducible module of

H_{K, {\bar{Q}}_{ℓ}}

, and let

H_{2 d}^{B M} ({\bar{S}}_{b}) [π_{f}] = {Hom}_{H_{K, \bar{Q}}^{ℓ}} (π_{f}, H_{2 d}^{B M} ({\bar{S}}_{b}) (- d)_{{\bar{Q}}_{ℓ}}) \otimes π_{f}

be the

π_{f}

-isotypical component. Then the cycle class map

c l : H_{2 d}^{B M} ({\bar{S}}_{b}) (- d) \to T_{ℓ}^{d} ({\bar{S}}_{K})

restricted to

H_{2 d}^{B M} ({\bar{S}}_{b}) [π_{f}]

is injective if the Satake parameter of

π_{f, p}

(the component of

π_{f}

p

) is

V_{μ}

-general.

(3) Assume that

{S h}_{K} (G)

is (essentially) a quaternionic Shimura variety or a Kottwitz arithmetic variety. Then the

π_{f}

-isotypical component of the cycle class map is surjective to

T_{ℓ}^{d} \bar{(S_{K})} [π_{f}]

if the Satake parameter of

π_{f, p}

is strongly

V_{μ}

-general. In particular, the Tate conjecture holds for these

π_{f}

Remark 3.3.2. (1) For a representation

V

\hat{G}

, the definitions of "

V

-general" and "strongly

V

-general" Satake parameters can be found in [64, DEFINITION 1.4.2]. Regular semisimple elements in

{^{c} G |}_{d = (p, σ_{p})}

are always

V

-general, but not the converse. See [64, remark 1.4.3].

(2) Some special cases of the theorem were originally proved in

[33, 60]

The proof of this theorem relies on several different ingredients. Via the RapoportZink uniformization of the basic locus of a Shimura variety, part (2) can be reduced a question about irreducible components of certain affine Deligne-Lusztig varieties, which was studied in [64, Â§3]. The most difficult is part (2), which we proved by calculating the intersection numbers among all

d

-dimensional cycles in

{\bar{S}}_{b}

. These numbers can be encoded in an

r \times r

matrix with entries in

H_{K_{p}}^{c l}

. In general, it seems hopeless to calculate this matrix directly and explicitly. However, this matrix can be understood as the composition of certain morphisms in

Coh ({Loc}_{c G, p}^{u r})

. Namely, first we realize

G^{'} (Q) ∖ G^{'} (A) / K

as a Shimura set with

μ^{'} = 0

its Shimura cocharacter. Then using Theorem 2.3.3 (and the Satake isomorphism (2.3)), this matrix can be calculated as

Then one needs to determine when this pairing is nondegenerate, which itself is an interesting question in representation theory, whose solution relies on the study of the Chevellay's restriction map for vector-valued functions. The determinant of this matrix was calculated in [65]. Finally, part (3) was proved by comparing two trace formulas, the Lefschetz trace formula for

G

and the Arthur-Selberg trace formula for

G^{'}

Example 3.3.3. Let

G = U (1, 2 r)

be the unitary group

^{7}

(2 r + 1)

-variables associated to an imaginary quadratic extension

E / Q

, whose signature is

(1, 2 r)

at infinity. It is equipped with a standard Shimura datum, giving a Shimura variety (after fixing a level

K \subset G (A_{f})

). In particular, if

r = 1

, this is (essentially) the Picard modular surface. Let

p

be a prime inert in

E

such that

K_{p}

is hyperspecial. In this case

{\bar{S}}_{b}

is a union of certain Deligne-Lusztig varieties, parametrized by

G^{'} (Q) ∖ G^{'} (A_{f}) / K

, where

G^{'} = U (0, 2 r + 1)

that is isomorphic to

G

at all finite places. The intersection patterns of these cycles inside

{\bar{S}}_{b}

were (essentially) given in [61] but the intersection numbers between these cycles are much harder to compute. In fact, we do not know how to compute them directly for general

r

, except applying Theorem 2.3.3 to this case. (The case

r = 1

can be handled directly.)

We have

\hat{G} = {G L}_{2 r + 1}

on which

σ_{p}

acts as

A \mapsto J {(A^{T})}^{- 1} J

, where

J

is the antidiagonal matrix with all entries along the antidiagonal being 1 . The representation

V_{μ}

is the standard representation of

{G L}_{2 r + 1}

. One checks that

\dim V_{μ}^{Tate} = 1

(which is consistent with the above mentioned parameterization of irreducible components of

{\bar{S}}_{b}

G^{'} (Q) ∖ G^{'} (A_{f}) / K)

. We identify the weight lattice of

\hat{G}

Z^{2 r + 1}

as usual. Then

{Hom}_{Coh ({Loc}_{c}^{u r}}^{u r} (O, \tilde{V_{μ}})

is a free rank one module over

{Hom}_{Coh ({Loc}_{C, p}^{u r}}^{u r} (O, O) = H_{K_{p}}^{c l} \otimes Q ℓ

. Then a generator

a_{in}

induces an

H_{K, Q_{ℓ}}

-equivariant homomorphism

S (a_{i n}) : C (G^{'} (Q) ∖ G^{'} (A_{f}) / K) \to H_{c}^{2 r} ({\bar{S}}_{K, {\bar{k}}_{v}}, Q_{ℓ} (r))

This is not an adjoint group so the example is not consistent with our assumption. But it is more convenient for the discussion here. The computations are essentially the same.
realizing the cycle class map of

{\bar{S}}_{b}

(up to a multiple). The module

{Hom}_{Coh (Loc}^{c_{G, p}}^{u r} (\tilde{V_{μ}}, O)

is also free of rank one over

H_{K_{p}, Q_{ℓ}}

. For a chosen generator

a_{out}

, the composition

S (a_{out}) \circ S (a_{i n}) = S (a_{o u t} \circ a_{i n})

calculates the intersection matrix of those cycles from the irreducible components of

{\bar{S}}_{b}

The element

h := a_{out} \circ a_{in} \in H_{K_{p}, Q_{ℓ}}

was explicitly computed in [65, EXAMPLE 6.4.2] (up to obvious modification and also via the Satake isomorphism (1.4)). Namely,

\begin{matrix} (3.3) & h = p^{r (r + 1)} \sum_{i = 0}^{r} (- 1)^{i} (2 i + 1) p^{(i - r) (r + i + 1)} \sum_{j = 0}^{r - i} {[\begin{matrix} 2 r + 1 - 2 j \\ r - i - j \end{matrix}]}_{t = - p} T_{p, j} \end{matrix}

Here,

T_{p, j} = 1_{K_{p} λ_{j} (p) K_{p}}

, with

λ_{i} = (1^{i}, 0^{2 r - 2 i + 1}, (- 1)^{i})

, and

{[\begin{array}{l} n \\ m \end{array}]}_{t}

is the

t

-analogue of the binomial coefficient given by

[0]_{t} = 1, [n]_{t} = \frac{t^{n} - 1}{t - 1}, [n]_{t}! = [n]_{t} [n - 1]_{t} \dots [1]_{t}, {[\begin{matrix} n \\ m \end{matrix}]}_{t} = \frac{[n]_{t}!}{[n - m]_{t}! [m]_{t}!}

In other words, the intersection matrix of cycles in

{\bar{S}}_{b}

in this case is calculated by the Hecke operator (3.3).

On interesting consequence is this computation is the following consequence on the intersection theory of the finite Deligne-Lusztig varieties, for which we do not know a direct proof. Let

W

be a

(2 r + 1)

-dimensional nondegenerate hermitian space over

F_{p^{2}}

. Consider the following

r

-dimensional Deligne-Lusztig variety

{D L}_{r} := {H \subset W of dimension r ∣ H \subseteq {(H^{(p)})}^{⊥}}

where

H^{(p)}

the pullback of

H

along the Frobenius. Let

H

denote the corresponding universal subbundle of rank

r

. Let

E = H^{(p)} \otimes ({(H^{(p)})}^{⊥} / H)

. Then we have

\begin{matrix} (3.4) & \int_{{D L}_{r}} c_{r} (E) = \sum_{i = 0}^{r} (- 1)^{i} (2 i + 1) p^{i^{2} + i} {[\begin{matrix} 2 r + 1 \\ r - i \end{matrix}]}_{t = - p} \end{matrix}

3.4. The Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives

Let

M

be a rational pure Chow motive of weight -1 over a number field

F

. The Beilinson-Bloch-Kato conjecture, which is a far reaching generalization of the Birch and Swinnerton-Dyer conjecture, predicts deep relations between certain algebraic, analytic, and cohomological invariants attached to

M

the rational Chow group $C H (M)^{0}$ of homologically trivial cycles of $M$ ;
the $L$ -function $L (s, M)$ of $M$ ;
the Bloch-Kato Selmer group $H_{f}^{1} (F, H_{ℓ} (M))$ of the $ℓ$ -adic realization $H_{ℓ} (M)$ of $M$ .

The Beilinson-Bloch conjecture predicts an equality

\dim_{Q} C H (M)^{0} = {ord}_{s = 0} L (s, M)

between the dimension of

C H (M)^{0}

and the vanishing order of the

L

-function at the central point, while the Bloch-Kato conjecture predicts

{ord}_{s = 0} L (s, M) = \dim_{Q_{ℓ}} H_{f}^{1} (F, H_{ℓ} (M))

In addition, the so-called

ℓ

-adic Abel-Jacobi map

{AJ}_{ℓ} : C H (M)^{0} \otimes Q_{ℓ} \to H_{f}^{1} (F, H_{ℓ} (M))

should be an isomorphism.

This conjecture seems to be completely out of reach at the moment. For example, for a general motive it is still widely open whether the

L

-function has a meromorphic continuation to the whole complex plane so that the vanishing order of

L (s, M)

s = 0

makes sense. (This would follow from the Galois-to-automorphic direction of the Langlands correspondence for number fields.) Despite this, there have been many works testing this conjecture in various special cases, mostly for motives

M

of small rank. In the work [49], we verify certain cases of the above conjecture for Rankin-Selberg motives, which consist of a sequence of motives of arbitrarily large rank.

We assume that

F / F^{+}

is a (nontrivial) CM extension with

F^{+}

totally real in the sequel.

Theorem 3.4.1. Let

A_{1}, A_{2}

be two elliptic curves over

F^{+}

. Assume that

(1)

{End}_{\bar{F}} A_{i} = Z

and

{Hom}_{\bar{F}} (A_{1}, A_{2}) = 0

;

(2)

{Sym}^{n - 1} A_{1}

and

{Sym}^{n} A_{2}

are modular;

(3)

F^{+} \neq Q

n \geq 3

Under these assumption, if

L (n, {Sym}^{n - 1} A_{1} \times {Sym}^{n} A_{2}) \neq 0

, then for almost

ℓ

\dim_{Q_{ℓ}} H_{f}^{1} (F, {Sym}^{n - 1} V_{ℓ} (A_{1}) \otimes {Sym}^{n} V_{ℓ} (A_{2}) (1 - n)) = 0

Here

V_{ℓ} (A_{i})

denotes the rational Tate module of

A_{i}

as usual.

This theorem is in fact a consequence of a more general result concerning BlochKato Selmer groups of Galois representations associated to certain Rankin-Selberg automorphic representations, which we now discuss.

Recall that for an irreducible regular algebraic conjugate self-dual cuspidal (RACSDC) automorphic representation

Π

{G L}_{n} (A_{F})

, one can attach a compatible system of Galois representations

ρ_{Π, λ} : Γ_{F} \to {GL}_{n} (E_{λ})

, where

E \subset C

is a large enough number field and

λ

is a prime of

E

(see [16]). Such

E

is called a strong coefficient field of

Π

, which in the situation considered below can be taken as the number field generated by Hecke eigenvalues of

Π

Theorem 3.4.2. Suppose that

F^{+} \neq Q

n \geq 3

. Let

Π_{n}

(resp.

Π_{n + 1})

be an RACSDC automorphic representation of

{G L}_{n} (A_{F})

(resp.

{G L}_{n + 1} (A_{F})

) with trivial infinitesimal character. Let

E \subseteq C

be a strong coefficient field for both

Π_{n}

and

Π_{n + 1}

. Let

λ

be an admissible prime of

E

with respect to

Π := Π_{0} \times Π_{1}

. Let

ρ_{Π, λ} := ρ_{Π_{n}, λ} \otimes_{E_{λ}} ρ_{Π_{n + 1}, λ}

(1) If the Rankin-Selberg

L

-value

L (\frac{1}{2}, Π) \neq 0^{8}

, then

H_{f}^{1} (F, ρ_{Π, λ} (n)) = 0

(2) If certain element

Δ_{λ} \in H_{f}^{1} (F, ρ_{Π, λ} (n))

(to be explained below) is non-zero, then

H_{f}^{1} (F, ρ_{Π, λ} (n))

is generated by

Δ_{λ}

as an

E_{λ}

-vector space.

The notion of admissible primes appearing in the theorem consists of a long list of assumptions, some of which are rather technical. Essentially, it guarantees that the Galois

and the reduction mod

λ

representation is suitably large and contains certain particular elements. (This is also related to the notion of

V

-general from Theorem 3.3.1.) Fortunately, in some favorable situations, one can show that all but finitely many primes are admissible. For example, this is the case considered in Theorem 3.4.1. For another case in pure automorphic setting, see

[49

, Ñ‚Ð½Ð¼. 1.1.7].

The proof of the theorem uses several different ingredients. The initial step for case (1) is to translate the analytic condition

L (\frac{1}{2}, Π) \neq 0

into a more algebraic condition via the global Gan-Gross-Prasad (GGP) conjecture. Namely, the GGP conjecture predicts that in this case, there exist a pair of hermitian spaces

(V_{n}, V_{n + 1})

over

F

that are totally positive definite at

\infty

, where

V_{n + 1} = V_{n} \oplus F v

with

(v, v) = 1

, and a tempered cuspidal automorphic representation

π = π_{n} \times π_{n + 1}

of the product of unitary groups

G = U (V_{n}) \times U (V_{n + 1})

, such that the period integral

[Δ_{H}] : C_{c}^{*} (Sh (G), E) [π] \to E

does not vanish, where

H := U (V_{n})

embeds into

G

diagonally, which induces an embedding

Δ_{H} : Sh (H) ↪ Sh (G)

of corresponding Shimura varieties (in fact, Shimura sets) with appropriately chosen level structures (here and later we omit level structures from the notations). We denote by

[Δ_{H}

] the homology class of

Sh (G)

given by

Sh (H)

and write

C_{c}^{*} (Sh (G), E) [π]

for the

π

-isotypical component of the cohomology (i.e., functions) of

Sh (G)

. This conjecture was first proved in [73] under some local restrictions which are too restrictive for arithmetic applications. Those restrictions are all lifted in our recent work through some new techniques in the study of trace formulae [8].

The strategy then is to construct, for every

m \geq 1

, (infinitely many) cohomology classes

{Θ_{m}^{p}}_{p} \subset H^{1} (F, {(ρ_{Π, λ} / λ^{m})}^{*} (1))

, where

p

are appropriately chosen primes and

(-)^{*} (1)

denotes the usual Pontryagin duality twisted by the cyclotomic character, such that the local Tate pairing at

p

between

Θ_{m}^{p}

and Selmer classes of the Galois representation

ρ_{Π, λ} / λ^{m}

is related to the above period integral. Then one can use Kolyvagin type argument (amplified in

[47, 49]

), with

{Θ_{m}^{p}}

served as annihilators of the Selmer groups, to conclude.

The construction of

Θ_{m}^{p}

uses the diagonal embedding of Shimura varieties

Δ_{H^{'}} : Sh (H^{'}) ↪ Sh (G^{'})

where

H^{'} ↪ G^{'}

are prime-to-

p

trivialized (extended pure) inner forms of

H \subset G

(see Definition 2.3.1). These Shimura varieties have parahoric level structures at

p

, and using

8 Here we use the automorphic normalization of the

L

-function.
the theory of local models (Section 3.1) one can show that their integral models are polysemistable at

p

and compute the sheaf of nearby cycles on their mod

p

fibers. Using many ingredients, including the understanding of (integral) cohomology of

Sh (G^{'})

over

\bar{F}

, the computations from Example 3.3.3 (in particular, (3.3) and (3.4)), and the Taylor-Wiles patching method [48], one shows that

{(ρ_{Π, λ} / λ^{m})}^{*} (1)

does appear in the cohomology of

Sh (G^{'})

(the so-called arithmetic level raising for

Π

), and that the diagonal cycle

Δ_{H^{'}}

, when localized at

{(ρ_{Π, λ} / λ^{m})}^{*} (1)

, does give the desired class

Θ_{m}^{p}

. We shall mention that this is consistent with conjectures in Sections 2.1 and 2.3, as coherent sheaves on

{Loc}_{G, p} \otimes O_{E} / λ^{m}

corresponding to

c

-ind

K_{p}^{G (Q_{p})} (O_{E} / λ^{m})

and

c

-ind

K_{p}^{'} G^{'} (Q_{p}) (O_{E} / λ^{m})

are expected to be related exactly in this way.

We could also explain the class

Δ_{λ}

appearing in case (2). Namely, in this case we start with an embedding of Shimura varieties

Δ_{H} : Sh (H) ↪ Sh (G)

, where

G

is a product of unitary groups such that

Π

descends to a tempered cuspidal automorphic representation

π

appearing in the middle dimensional cohomology of

{S h}_{G}

. Then the

π

-isotypical component of the cycle

Δ_{H}

is homologous to zero, and we let

Δ_{λ} = {AJ}_{λ} (Δ_{H} [π])

. The strategy to prove case (2) then is to reduce it to case (1) via some similar arguments as before.

ACKNOWLEDGMENTS

The author would like to thank all of his collaborators, without whom many works reported in this survey article would not be possible. Besides, he would also like to thank Edward Frenkel, Dennis Gaitsgory, Xuhua He, Michael Rapoport, Peter Scholze for teaching him and for discussions on various parts of the Langlands program over years.

FUNDING

The work is partially supported by NSF under agreement No. DMS-1902239 and a Simons fellowship.

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XINWEN ZHU

Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA, xzhu@caltech.edu

4. ALGEBRAIC AND COMPLEX GEOMETRY

MARC LEVINE

ABSTRACT

We give a survey of the development of motivic cohomology, motivic categories, and some of their recent descendants.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 14F42; Secondary 14C15, 19D55, 19D45, 55P42

KEYWORDS

Motivic cohomology, motives,

K

-theory, algebraic cycles, motivic homotopy theory

1. INTRODUCTION

Motivic cohomology arose out of a marriage of Grothendieck's ideas about motives with a circle of conjectures about special values of zeta functions and

L

-functions. It has since taken on a very active life of its own, spawning a multitude of developments and applications. My intention in this survey is to present some of the history of motivic cohomology and the framework that supports it, its current state, and some thoughts about its future directions. I will say very little about the initial impetus given by the conjectures about zeta functions and

L

-functions, as there are many others who are much better qualified to tell that story. I will also say next to nothing about the many applications motivic cohomology has seen: I think this would be like writing about the applications of cohomology up to, say, 1950, and would certainly make this already lengthy survey completely unmanageable.

My basic premise is that motivic cohomology is supposed to be universal cohomology for algebro-geometric objects. As "universal" depends on the universe one happens to find oneself in, motivic cohomology is an ever-evolving construct. My plan is to give a path through some of the various universes that have given rise to motivic cohomologies, to describe the resulting motivic cohomologies and put them in a larger, usually categorical, framework. Our path will branch into several directions, reflecting the different aspects of algebraic and arithmetic geometry that have been touched by this theory. We begin with the conjectures of Beilinson and Lichtenbaum about motivic complexes that give rise to the universal Bloch-Ogus cohomology theory on smooth varieties over a field, and the candidate complexes constructed by Bloch and Suslin. We then take up Voevodsky's triangulated category of motives over a field and the embedding of the motivic complexes and motivic cohomology in this framework. The next developments moving further in this direction give us motivic homotopy categories that tell us about "generalized motivic cohomology" for a much wider class of schemes, analogous to the development of the stable homotopy category and generalized cohomology for spaces; this includes a number of candidate theories for motivic cohomology over a general base-scheme. We conclude with three variations on our theme:

Milnor-Witt motives and Milnor-Witt motivic cohomology, incorporating information about quadratic forms,
Motives with modulus, relaxing the usual condition of homotopy invariance with respect to the affine line, and
$p$ -adic, Ã©tale motivic cohomology in mixed characteristic $(0, p)$ , with its connection to $p$ -adic Hodge theory.

This last example does not yet, as far as I know, have a categorical framework, while one for a motivic cohomology with modulus is still in development.

There is already an extensive literature on the early development of motives and motivic cohomology. It was not my intention here to cover this part in detail, but I include a section on this topic to give a quick overview for the sake of background, and to isolate a few main ideas so the reader could see how they have influenced later developments.

I would like to thank all those who helped me prepare this survey, especially Tom Bachmann, Federico Binda, Dustin Clausen, Thomas Geisser, Wataru Kai, Akhil Mathew, Hiroyasu Miyazaki, Matthew Morrow, and Shuji Saito. In spite of their efforts, I feel certain that a number of errors have crept in, which are, of course, all my responsibility. I hope that the reader will be able to repair them and continue on.

2. BACKGROUND AND HISTORY

2.1. The conjectures of Beilinson and Lichtenbaum

Beilinson pointed out in his 1983 paper "Higher regulators and values of

L

-functions" [13] that the existence of Gillet's Chern character [53] from algebraic

K

-theory to an arbitrary Bloch-Ogus cohomology theory [30] with coefficients in a

Q

-algebra implies that one can form the universal Bloch-Ogus cohomology

H_{μ}^{a} (-, Q (b))

with

Q

-coefficients by decomposing algebraic

K

-theory into its weight spaces for the Adams operations

ψ_{k}

. In terms of the indexing, one has

H^{a} (X, Q (b)) := K_{2 b - a} (X)^{(b)}

where

K_{2 b - a} (X)^{(b)} \subset K_{2 b - a} (X)_{Q}

is the weight

b

eigenspace for the Adams operations

K_{2 b - a} (X)^{(b)} := {x \in K_{2 b - a} (X)_{Q} ∣ ψ_{k} (x) = k^{b} \cdot x}

This raised the question of finding the universal integral Bloch-Ogus cohomology theory. Let

{S c h}_{k}

denote the category of separated finite-type

k

-schemes with full subcategory

{S m}_{k}

of smooth

k

-schemes. Beilinson [13] and Lichtenbaum [87] independently conjectured that this universal theory

H_{μ}^{a} (-, Z (b))

should arise as the hypercohomology of a complex of sheaves

X \mapsto Γ_{X} (b)

{S m}_{k}

H_{μ}^{a} (X, Z (b)) := H^{a} (X, Γ_{X} (b))

with the

Γ_{X} (b)

satisfying a number of axioms. We give Beilinson's list of axioms for motivic complexes in the Zariski topology (axiom iv

(p)

was added by Milne [90, $2]):

(i) In the derived category of sheaves on

X, Γ (0)

is the constant sheaf

Z

{S m}_{k}

Γ (1) = G_{m} [- 1]

and

Γ (n) = 0

for

n < 0

(ii) The graded object

Γ (*) := [X \mapsto ⨁_{n \geq 0} Γ_{X} (n)]

is a commutative graded ring in the derived category of sheaves on

{S m}_{k}

(iii) The cohomology sheaves

H^{m} (Γ (n))

are zero for

m > n

and for

m \leq 0

n > 0

;

H^{n} (Γ (n))

is the sheaf of Milnor

K

-groups

X \mapsto K_{n, X}^{M}

(iv)(a) Letting

α : {S m}_{k, ét} \to {S m}_{k, Z a r}

Γ (n)_{ét} := α^{*} Γ (n)

Γ (n)

satisfies

Γ (n)_{ett} / m ≅ μ_{m}^{\otimes n}

for

m

prime to the characteristic, where

μ_{m}

m

th roots of unity.

(iv)(b) For

m

prime to the characteristic, the natural map

Γ (n) / m \to R α_{*} Γ (n)_{ett} / m

induces an isomorphism

Γ (n) / m \to τ_{\leq n} R α_{*} Γ (n)_{et} / m

. Integrally,

Z (n) \to R α_{*} Γ (n)_{ét}

induces an isomorphism

Γ (n) \to τ_{\leq n} R α_{*} Γ (n)_{ét}

and

R^{n + 1} α_{*} Γ (n)_{ét} = 0

(iv)(

p

) For

k

of characteristic

p > 0

, let

W_{v} Ω_{log}^{n}

denote the

ν

-truncated logarithmic de Rham-Witt sheaf. The

d \log

map

d \log : K_{n}^{M} / p^{n} \to W_{ν} Ω_{\log}^{n}

induces via (ii) a map

Γ (n) / p^{ν} \to W_{v} Ω_{\log}^{n} [- n]

, which is an isomorphism.

One then defines motivic cohomology by

H^{p} (X, Z (q)) := H^{p} (X_{Z a r}, Γ_{X} (q))

(v) There should also be a spectral sequence starting with integral motivic cohomology and converging to algebraic

K

-theory, analogous to the Atiyah-Hirzebruch spectral sequence from singular cohomology to topological

K

-theory. Explicitly, this should be

E_{2}^{p, q} := H^{p - q} (X, Z (- q)) \Rightarrow K_{- p - q} (X)

This spectral sequence should degenerate rationally, and give an isomorphism

H^{p} (X, Q (q)) := H^{p} (X, Z (q)) \otimes_{Z} Q ≅ K_{2 q - p} (X)^{(q)}

The vanishing

H^{m} (Γ (n)) = 0

for

n > 0

and

m \leq 0

m

-part of axiom (iv)(b) is known as the Beilinson-Lichtenbaum conjecture; this implies the integral part of (iv)(b) with the exception of the vanishing of

R^{n + 1} α_{*} Γ (n)_{ét}

, which is known as Hilbert's theorem 90 for the motivic complexes. In weight

n = 1

, with the identity

Γ (1) = G_{m} [- 1]

, this translates into the classical Hilbert theorem 90

H_{e t}^{1} (O, G_{m}) = 0

for

O

a local ring, while the mod

m

1 \to μ_{m} \to G_{m} \overset{\times m}{\to} G_{m} \to 1

In light of axiom (iii), the Merkurjev-Suslin theorem [89, Ñ‚HEOREM 14.1] settled the degree

\geq 2

part of (iv)(b) for

n = 2

even before the complex

Γ

(2) was defined.

Beilinson [14, $5.10] rephrased and refined these conjectures to a categorical statement, invoking a conjectural category of mixed motivic sheaves, and an embedding of the hypercohomology of the Beilinson-Lichtenbaum complexes into a categorical framework.

In this framework, motivic cohomology should arise via an abelian tensor category of motivic sheaves on

{Sch}_{S}, X \mapsto {Sh}^{mot} (X)

, admitting the six functor formalism of Grothendieck,

f^{*}, f_{*}, f_{!}, f^{!}, H o m, \otimes

, on the derived categories. There should be Tate objects

Z_{X} (n) \in {Sh}^{mot} (X)

, and objects

M (X) := p_{X!} p_{X}^{!} Z_{S} (0)

in the derived category of

{Sh}^{m o t} (S), p_{X} : X \to S

the structure morphism, and motivic cohomology should arise as the Hom-groups

H_{μ}^{a} (X, Z (b)) = {Hom}_{D ({S h}^{m o t} (S))} (M (X), Z_{S} (b) [a])

For

X

smooth over

S

, this gives the identity

H_{μ}^{a} (X, Z (b)) = {Ext}_{{S h}^{mot} (X)}^{a} (Z_{X} (0), Z_{X} (b))

This is a very strong statement, with implications that have not been verified to this day. For instance, the vanishing of

{Ext}_{A}^{a} (-, -)

for an abelian category

A

and for

a < 0

gives a vanishing

H^{m} (Z_{S} (n)) = 0

for

m < 0

. The stronger vanishing posited by axiom (iii) above (with

Q

n = 1

(for which the strong version holds).

Beilinson's conjecture on categories of motivic sheaves is still an open problem. However, other than the integral Beilinson-SoulÃ© vanishing conjecture, the axioms do not rely on the existence of an abelian category of motivic sheaves, and can be framed in the setting of a functorial assignment

X \mapsto D M (X)

from

S

-schemes to tensor-triangulated categories. Such a functor has been constructed and the axioms (except for the vanishing conjectures) have been verified. We will discuss this construction in Section 2.4.

2.2. Bloch's higher Chow groups and Suslin homology

The first good definition of motivic cohomology complexes was given by Spencer Bloch, in his landmark 1985 paper "Algebraic cycles and higher Chow groups" [24]. As

cycles modulo rational equivalence.

For

X

a finite type

k

-scheme, recall that the group of dimension

d

algebraic cycles on

X, Z_{d} (X)

, is the free abelian group on the integral closed subschemes

Z

X

of dimension

d

over

k

. The group of cycles modulo rational equivalence,

{C H}_{d} (X)

, has the following presentation. Let

n \mapsto Δ^{n}

be the cosimplicial scheme of algebraic

n

-simplices

Δ^{n} := Spec Z [t_{0}, \dots, t_{n}] / \sum_{i = 0}^{n} t_{i} - 1 ≅ A_{Z}^{n}

The coface and codegeneracy maps are defined just as for the usual real simplices

Δ_{top}^{n} \subset R^{n}

. A face of

Δ^{n}

is a closed subscheme defined by the vanishing of some of the

t_{i}

. Let

z_{d} (X, n)

be the subgroup of the

(n + d)

-dimensional algebraic cycles

Z_{n + d} (X \times Δ^{n})

generated by the integral closed

W \subset X \times Δ^{n}

such that

\dim W \cap X \times F = m + d

for each

m

-dimensional face

F

(or the intersection is empty). For cycles

w \in z_{d} (X, n)

, the face condition gives a well-defined pullback

{({Id}_{X} \times g)}^{*} : z_{d} (X, n) \to z_{d} (X, m)

for each map

g : Δ^{m} \to Δ^{n}

in the cosimplicial structure, forming the simplicial abelian group

n \mapsto z_{d} (X, n)

and giving the associated chain complex

z_{d} (X, *)

, Bloch's cycle complex. The degree 0 and 1 terms of

z_{d} (X, *)

give our promised presentation of

{C H}_{d} (X)

H_{0} (z_{d} (X, *)) = {C H}_{d} (X)

and Bloch defines his higher Chow group

{C H}_{d} (X, n)

{C H}_{d} (X, n) := H_{n} (z_{d} (X, *))

X

has pure dimension

N

over

k

, we index by codimension

z^{q} (X, *) := z_{N - q} (X, *); {C H}^{q} (X, n) := {C H}_{N - q} (X, n)

With some technical difficulties due to the necessity of invoking moving lemmas to allow for pullback morphisms, the assignment

X \mapsto z^{q} (X, 2 q - *)

can be modified via isomorphisms in the derived category to a presheaf of cohomological complexes

Z_{B l} (q)

{S m}_{k}

Following a long series of works

[25, 29, 43, 52, 94, 96, 112, 113, 115 - 117, 120, 121, 123 - 125

, 127] (see also [56, 102] for detailed discussions of the Bloch-Kato conjecture, the essential point in axiom (iv)(b) and the most difficult of the Beilinson axioms to prove), it has been shown that the complexes

Z_{B I} (q)

After Bloch introduced his cycle complexes, Suslin [111] constructed an algebraic version of singular homology. For a

k

-scheme

X

, instead of a naive generalization of the singular chain complex of a topological space by taking the free abelian group on the morphisms

Δ_{k}^{n} \to X

, Suslin's insight was to enlarge this to the abelian group of finite correspondences.

A subvariety

W

of a product

Y \times X

of varieties (with

Y

smooth) defines an irreducible finite correspondence from

Y

X

p_{1} : W \to Y

is finite and surjective to some irreducible component of

Y

. The association

y \mapsto p_{2} (p_{1}^{- 1} (y))

can be thought of as a multivalued map from

Y

X

The group of finite correspondences

{Cor}_{k} (Y, X)

is defined as the free abelian group on the irreducible finite correspondences. Given a morphism

f : Y^{'} \to Y

, there is a pullback map

f^{*} : {Cor}_{k} (Y, X) \to {Cor}_{k} (Y^{'}, X)

, compatible with the interpretation as multivalued functions, and making

{Cor}_{k} (-, X)

into a contravariant functor from smooth varieties over

k

to abelian groups.

Suslin defines

C_{n}^{Sus} (X) := {Cor}_{k} (Δ_{k}^{n}, X)

; the structure of

Δ_{k}^{*}

as smooth cosimplicial scheme makes

n \mapsto C_{n}^{Sus} (X)

a simplicial abelian group. As above, we have the associated complex

C_{*}^{Sus} (X)

, the Suslin complex of

X

, whose homology is the Suslin homology of

X

H_{n}^{S u s} (X, Z) := π_{n} (| m \mapsto C_{m}^{S u s} (X) |) = H_{n} (C_{*}^{S u s} (X))

In fact, the monoid of the

N

-linear combinations of irreducible correspondences

W \subset X \times Y

is the same as the monoid of morphisms

ϕ : X \to ⨆_{n \geq 0} {Sym}^{n} Y

where

{Sym}^{n} Y

is the quotient

Y^{n} / Σ_{n}

Y^{n}

by the symmetric group permuting the factors, with the monoid structure induced by the sum map

{Sym}^{n} Y \times {Sym}^{m} Y \to {Sym}^{m + n} Y

Suslin's complex and his definition of algebraic homology can thus be thought of as an algebraic incarnation of the theorem of Dold-Thom [34, sATz 6.4], that identifies the homotopy groups of the infinite symmetric product of a pointed

C W

complex

T

with the
reduced homology of

T

. The main result of [112] gives an isomorphism of the mod

n

Suslin homology,

H_{*}^{Sus} (X, Z / n)

, for

X

of finite type over

C

, with the

mod n

singular homology of

X (C)

, a first major success of the theory.

Let

Δ_{top}^{n}

denote the usual

n

-simplex

Δ_{top}^{n} := {(t_{0}, \dots, t_{n}) \in R^{n + 1} ∣ \sum_{i} t_{i} = 1, t_{i} \geq 0}

with the inclusion

Δ_{top}^{n} \subset Δ^{n} (C)

Theorem 2.1 ([112, THEOREM 8.3]). Let

X

be separated finite type scheme over

C

and let

n \geq 2

be an integer. Then the map

Hom (Δ_{C}^{*}, ⨆_{d \geq 0} {Sym}^{d} X) \to {Hom}_{top} (Δ_{top}^{*}, ⨆_{d \geq 0} {Sym}^{d} X (C))

induced by the inclusions

Δ_{top}^{m} \subset Δ^{m} (C)

gives rise to an isomorphism

H_{*}^{Sus} (X, Z / n) \to

H_{*}^{sing} (X (C), Z / n)

There is also a corresponding statement for

X

over an arbitrary algebraically closed field

k

p > 0

and

n

prime to

p

by using alterations.

2.3. Quillen-Lichtenbaum conjectures

Quillen's computation of the higher algebraic

K

-theory of finite fields and of number rings led to a search for a relation of higher algebraic

K

-theory with special values of zeta-functions and

L

-functions. We will not go into this in detail here, but to large part, this was responsible for the Beilinson-Lichtenbaum conjectures on the existence of motivic complexes computing the conjectural motivic cohomology. Going back to

K

-theory, this suggested that algebraic

K

-theory with mod-

ℓ

coefficients should be the same as mod-

ℓ

Ã©tale

K

-theory (a purely algebraic version of mod-

ℓ

topological

K

-theory, see [35]), at least in large enough degrees. This is more precisely stated as the Quillen-Lichtenbaum conjecture

Conjecture 2.2 ([101], [42, CONJEcture 3.9]). Let

ℓ

be a prime and let

X

be a regular, noetherian scheme with

ℓ

invertible on

X

. Suppose

X

has finite

ℓ

{cd}_{ℓ} (X)

. Then the canonical map

K_{n} (X; Z / ℓ^{r}) \to K_{n}^{e ́ t} (X; Z / ℓ^{r})

is an isomorphism for

n \geq {cd}_{ℓ} (X) - 1

and is injective for

n = {cd}_{ℓ} (X) - 2

Here

K_{n}^{et} (X; Z / ℓ)

is the Ã©tale

K

-theory developed by Dwyer and Friedlander [35, 41,42].

Conjecture 2.2 for a smooth

k

-scheme is essentially a consequence of the BeilinsonLichtenbaum axioms (without Beilinson-SoulÃ© vanishing). The Beilinson-Lichtenbaum conjecture (iv)(a,b) says that the comparison map

Γ (q) / ℓ^{r} \to R α_{*} μ_{ℓ^{r}}^{\otimes q}

induces an isomorphism on cohomology sheaves up to degree

q

. Combining the local-global spectral sequence
for some

X \in {S m}_{k}

with the Atiyah-Hirzebruch spectral sequences from motivic cohomology to

K

K

-theory, and keeping track of the cohomological bound in the Beilinson-Lichtenbaum conjecture gives the result.

2.4. Voevodsky's category DM and modern motivic cohomology

One can almost realize Beilinson's ideas of a categorical framework for motivic cohomology by working in the setting of triangulated categories, viewed as a replacement for the derived category of Beilinson's conjectured abelian category of motivic sheaves. Once this is accomplished, one could hope that an abelian category of mixed motives could be constructed out of the triangulated category as the heart of a suitable

t

-structure.

Constructions of a triangulated category of mixed motives over a perfect base-field were given by Hanamura [57-59], Voevodsky [127], and myself [83]. All three categories yield Bloch's higher Chow groups as the categorical motivic cohomology, however, Voevodsky's sheaf-theoretic approach has had the most far-reaching consequences and has been widely adopted as the correct solution. The construction of a motivic

t

-structure is still an open problem.

^{1}

There are also constructions of triangulated categories of mixed motives by the method of compatible realizations, such as by Huber [64], or Nori's construction of an abelian category of mixed motives, described in [65, PART II]; we will not pursue these directions here. We also refer the reader to Jannsen's survey on mixed motives [68].

Voevodsky's triangulated category of motives over

k, DM (k)

, is based on the category of finite correspondences on

{S m}_{k}

, a refinement of Grothendieck's composition law for correspondences on smooth projective varieties. Grothendieck had constructed categories of motives for smooth projective varieties, with the morphisms from

X

Y

given by the group of cycles modulo rational equivalence

{C H}_{\dim X} (X \times Y)

. The composition law is given by

\begin{matrix} (2.1) & W^{'} \circ W = p_{X Z *} (p_{X Y}^{*} (W) \cdot p_{Y Z}^{*} (W^{'})) \end{matrix}

with

W \in {C H}_{\dim X} (X \times Y)

and

W^{'} \in {C H}_{\dim Y} (Y \times Z)

; one needs to pass to cycle classes to define

p_{X Y}^{*} (W) \cdot p_{Y Z}^{*} (W^{'})

and the projection

p_{X Z}

needs to be proper (that is,

Y

needs to be proper over

k

) to define

p_{X Z *}

Voevodsky's key insight was to restrict to finite correspondences, so that all the operations used in the composition law of correspondence classes would be defined on the level of the cycles themselves, without needing to pass to rational equivalence classes, and without needing the varieties involved to be proper. Voevodsky's idea of having a well-defined composition law on a restricted class of correspondences has been modified and applied in a wide range of different contexts, somewhat similar to the use of various flavors of bordism theories in topology.

Let

X

and

Y

be in

{Sm}_{k}

. Recall from Section 2.2 the subgroup

{Cor}_{k} (X, Y) \subset

Z_{\dim X} (X \times Y)

generated by the integral

W \subset X \times Y

that are finite over

X

and map surjectively to a component of

X

Lemma 2.3. Let

X, Y, Z

be smooth

k

-varieties and take

α \in {Cor}_{k} (X, Y), β \in {Cor}_{k} (Y, Z)

. Then

(i) The cycles

p_{Y Z}^{*} (β)

and

p_{X Y}^{*} (α)

intersect properly on

X \times Y \times Z

, so the inter section product

p_{Y Z}^{*} (β) \cdot p_{X Y}^{*} (α)

exists as a well-defined cycle on

X \times Y \times Z

(ii) Letting

| α | \subset X \times Y

, and

| β | \subset Y \times Z

denote the support of

α

and

β

, respectively, each irreducible component of the intersection

X \times | β | \cap | α | \times Z

is finite over

X \times Z

, and maps surjectively onto some component of

X

In other words, the formula

β \circ α = p_{X Z *} (p_{Y Z}^{*} β \cdot p_{X Y}^{*} α)

makes sense for

α \in {Cor}_{k} (X, Y)

and

β \in {Cor}_{k} (Y, Z)

, and the resulting cycle on

X \times Z

is in

{Cor}_{k} (X, Z)

. This defines the composition law in Voevodsky's category of finite correspondences,

{Cor}_{k}

, with objects as for

{Sm}_{k}

, and morphisms

{Hom}_{{Cor}_{k}} (X, Y) = {Cor}_{k} (X, Y)

. Sending a usual morphism

f : X \to Y

of smooth varieties to its graph defines a faithful functor

[-] : {S m}_{k} \to

Cor

_{k}

Once one has the category

{C o r}_{k}

, the path to

D M (k)

is easy to describe. One takes the category of additive presheaves of abelian groups on

{C o r}_{k}

, the category of presheaves with transfer

PST (k)

. Inside

PST (k)

is the category NST

(k)

of Nisnevich sheaves with transfer, that is, a presheaf that is a Nisnevich sheaf when restricted to

{Sm}_{k} \subset {Cor}_{k}

. Each

X \in {Sm}_{k}

defines an object

Z_{t r} (X) \in NST (k)

, as the representable (pre)sheaf

Y \mapsto {Cor}_{k} (Y, X)

. Inside the derived category

D (NST (k))

is the full subcategory of complexes

K

whose homology presheaves

{\underline{h}}_{i} (K)

are

A^{1}

-homotopy invariant:

{\underline{h}}_{i} (K) (X) ≅ {\underline{h}}_{i} (K) (X \times A^{1})

for all

X \in {S m}_{k}

. This is the category of effective motives

{D M}^{eff} (k)

. The Suslin complex construction,

P \mapsto C_{*}^{Sus} (P)

, with

C_{*}^{S u s} (P) (X) := P (X \times Δ^{*})

extends to a functor

R C_{*}^{Sus} : D (N S T (k)) \to {D M}^{eff} (k)

, and realizes

{D M}^{eff} (k)

as the localization of

D (NST (k))

with respect to the complexes

Z_{t r} (X \times A^{1}) \overset{p_{*}}{\to} Z_{t r} (X)

. Via

R C_{*}^{Sus}

{D M}^{e f f} (k)

inherits a tensor structure

\otimes

from

D (N S T (k))

. The functor

Z_{t r} : {S m}_{k} \to N S T (k)

defines the functor

M^{eff} := R C_{*}^{S u s} \circ Z_{t r}

M^{e f f} : {S m}_{k} \to {D M}^{e f f} (k)

The Tate object

Z (1) \in {D M}^{e f f} (k)

is the image of the complex

Z_{t r} (

Spec

k) \overset{i_{\infty *}}{\to}

Z_{t r} (P^{1})

(with

Z_{t r} (P^{1})

in degree 2 ) via

R C_{*}^{Sus}

. One forms the triangulated tensor category

DM (k)

as the category of

- \otimes Z (1)

-spectrum objects in

{D M}^{eff} (k)

, inverting the endofunctor

- \otimes Z (1)

; for

M \in D M (k)

, one has the Tate twists

M (n) := M \otimes Z (1)^{\otimes n}

for

n \in Z

; in particular, we have the Tate objects

Z (n)

. The functor

M^{eff}

induces the functor

M : {S m}_{k} \to D M (k)

Bloch's higher Chow groups, Suslin homology, and the motivic complexes

Z_{B I} (q)

are represented in

D M (k)

via canonical isomorphisms

\begin{aligned} {C H}^{q} (X, 2 q - p) & = H^{p} (X_{Z a r}, Z_{B l} (q)) ≅ {Hom}_{D M (k)} (M (X), Z (q) [p]) \\ H_{n}^{S u s} (X, Z) & = H_{n} (C_{*}^{S u s} (X)) ≅ {Hom}_{D M (k)} (Z [n], M (X)) \end{aligned}

In addition, one has the presheaf of complexes

Z_{V} (q)

{S m}_{k}

Z_{V} (q) (X) := C_{- *}^{S u s} (Z_{t r} {(G_{m})}^{\otimes q} [- q]) (X)

where

Z_{t r} (G_{m})

is the quotient presheaf

Z_{t r} (A^{1} ∖ {0}) / Z_{t r} ({1})

. The complexes

Z_{V} (q)

and

Z_{B I} (q)

define isomorphic objects in

{D M}^{e f f} (k)

, in particular, are isomorphic in the derived category of Nisnevich sheaves on

{S m}_{k}

. The details of these constructions and results are carried out in [127] (with a bit of help from [117]).

2.5. Motivic homotopy theory

Although Voevodsky's triangulated category of motives does give motivic cohomology a categorical foundation, this is really a halfway station on the way to a really suitable categorical framework. As analogy, embedding the Beilinson-Lichtenbaum/Bloch-Suslin theory of motivic complexes in

D M (k)

is like considering the singular chain or cochain complex of a topological space as an object in the derived category of abelian groups. A much more fruitful framework for singular (co)homology is to be found in the stable homotopy category SH.

A parallel representability for motivic cohomology for schemes over a base-scheme

B

in a wider category of good cohomology theories is to be found in the motivic stable homotopy category over

B, S H (B)

. This, together with the motivic unstable homotopy category,

H (B)

, gives the proper setting for the deeper study of motivic cohomology, besides placing this theory on a equal footing with all cohomology theories on algebraic varieties that satisfy a few natural axioms.

Just as the category

D M (k)

starts out as a category of presheaves, the category

S H (B)

starts out with the category of presheaves of simplicial sets on

{S m}_{B}

. The construction of the unstable motivic homotopy category

H (B)

over a general base-scheme

B

as a suitable localization of this presheaf category was achieved by Morel-Voevodsky [94] and the stable version

S H (B)

was described by Voevodsky in his ICM address [116]. The important sixfunctor formalism of Grothendieck was sketched out by Voevodsky and realized in detail by Ayoub

[5, 6]

. A general theory of motivic categories with a six-functor formalism, including

S H (-)

, was established by Cisinski-DÃ©glise [33], and Hoyois [62] gave a construction on the level of infinity categories for an equivariant version. A new point of view, the approach of framed correspondences, also first sketched by Voevodsky [126], is a breakthrough in our understanding of the infinite loop objects in the motivic setting, and concerning our main interest, motivic cohomology, has led to a natural construction of motivic cohomology over a general base-scheme.

In topology, the representation of singular (co)homology via the singular (co)chain complexes is placed in the setting of stable homotopy theory through the construction of the

Eilenberg-MacLane spectra, giving a natural isomorphism for each abelian group

A

H^{n} (X, A) ≅ {Hom}_{S H} (Σ^{\infty} X_{+}, Σ^{n} EM (A))

with the Eilenberg-MacLane spectrum

EM (A)

being characterized by its stable homotopy groups

π_{n}^{s} (EM (A)) = {\begin{cases} A & for n = 0 \\ 0 & else \end{cases}

The assignment

A \mapsto EM (A)

extends to a fully faithful embedding EM :

D (A b) \to S H

. This realizes the ordinary (co)homology as being represented by the derived category

D (A b)

via its Eilenberg-MacLane embedding in SH, which in turn is to be viewed as the category of all cohomology theories on reasonable topological spaces.

The stable homotopy category SH is the stabilization of the unstable pointed homotopy category

H_{∙}

with respect to the suspension operator

Σ X := S^{1} \land X

, which becomes an invertible endofunctor on

S H

. The resulting functor of

H_{∙}

to its stabilization is the infinite suspension functor

Σ^{\infty}

and gives us the "effective" subcategory

{S H}^{eff} \subset S H

, as the smallest subcategory containing

Σ^{\infty} (H_{∙})

and closed under homotopy cofibers and small coproducts. This in turn gives a decreasing filtration on

S H

by the subcategories

Σ^{n} {S H}^{e f f}

n \in Z

. This rather abstract looking filtration is simply the filtration by connectivity:

E

is in

Σ^{n} {S H}^{e f f}

if and only if

π_{m}^{s} E = 0

for

m < n

. The layers in this filtration are isomorphic to the category

A b

, by the functor

E \mapsto π_{n} E

, and in fact, this filtration is the one given by a natural

t

-structure on

S H

with heart

A b

; concretely, the 0 th truncation

τ_{0} E

is given by the Eilenberg-MacLane spectrum

EM (π_{0} (E))

A central example is the sphere spectrum

S := Σ^{\infty} S^{0}

. Since

π_{0}^{s} S = {colim}_{m} π_{m} (S^{m}) = Z

we have

τ_{0} S = EM (Z)

, establishing the natural relation between homology and homotopy.

In the motivic world, we have a somewhat parallel picture. The pointed unstable category

H_{∙} (B)

has a natural 2-parameter family of "spheres." Let

S^{n}

denote the constant presheaf with value the pointed

n

-sphere, and let

G_{m}

denote the representable presheaf

A^{1} ∖ {0}

pointed by 1 . Define

S^{a, b} := S^{a - b} \land G_{m}^{\land b}

for

a \geq b \geq 0

. We consider

P^{1}

as the representable presheaf, pointed by 1 ; there is a canonical isomorphism

P^{1} ≅ S^{2, 1}

H_{∙} (B)

In order to achieve the analog of Spanier-Whitehead duality in the motivic setting, one needs to use spectra with respect to

P^{1}

-suspension rather than with respect to

S^{1}

-suspension. The category

S H (B)

is constructed as a homotopy category of

P^{1}

-spectra in

H_{∙} (B)

, so

P^{1}

-suspension becomes invertible and our family of spheres extends to a family of invertible suspension endofunctors

Σ^{a, b} : S H (B) \to S H (B), a, b \in Z

Each

E \in S H (B)

gives the bigraded cohomology theory on

{S m}_{B}

E^{a, b} (X) := {Hom}_{S H (B)} (Σ_{P^{1}}^{\infty} X_{+}, Σ^{a, b} \land E)

Note that the translation in

S H (B)

is given by

S^{1}

-suspension, not

P^{1}

-suspension.

The effective subcategory

{S H}^{eff} (B)

is defined as the localizing subcategory (i.e., a triangulated subcategory closed under small coproducts) generated by the

P^{1}

-infinite suspension spectra

Σ_{P^{1}}^{\infty} X

for

X \in H_{∙} (B)

. We replace the filtration of

S H

with respect to

S^{1}

-connectivity with the filtration on

S H (B)

with respect to

P^{1}

-connectivity, via the subcategories

Σ_{P_{1}^{1}}^{n} {S H}^{e f f} (B)

. This is Voevodsky's slice filtration, with associated

n

th truncation denoted

f_{n}

, giving for each

E \in S H (B)

the tower

\dots \to f_{n + 1} E \to f_{n} E \to \dots \to E

One has the layers

s_{n} E

of this tower, fitting into a distinguished triangle

f_{n + 1} E \to f_{n} E \to s_{n} E \to f_{n + 1} E [1] = Σ^{1, 0} f_{n + 1} E

An important difference from the topological case is that this is a filtration by triangulated subcategories; the

P^{1}

-suspension is not the shift in the triangulated structure on

S H (B)

, and so the slice filtration does not arise from a

t

-structure.

We concentrate for a while on the case

B = Spec k, k

a perfect field. There is an Eilenberg-MacLane functor

E M : DM (k) \to S H (k)

giving the motivic cohomology spectrum

EM (Z (0)) \in S H (k)

representing motivic cohomology as

H^{p} (X, Z (q)) = EM (Z (0))^{p, q} (X)

One has the beautiful internal description of motivic cohomology via Voevodsky's isomorphism [122]

\begin{matrix} (2.2) & s_{0} S_{k} ≅ EM (Z (0)) \end{matrix}

see also [85, THEOREM 10.5.1] and the recent paper of Bachmann-Elmanto [9]. In other words, the 0th slice truncation of the motivic sphere spectrum represents motivic cohomology. RÃ¶ndigs-Ã˜stvÃ¦r [103] show that the homotopy category of

EM (Z (0))

-modules in

S H (k)

is equivalent to

D M (k)

and represents the Eilenberg-MacLane functor as the forgetful functor, right-adjoint to the free

EM (Z (0))

functor

DM (k)

This is the triangulated motivic analog of the classical result, that the heart of the

t

-structure on

S H

A b

2.6. Motivic cohomology and the rational motivic stable homotopy category

In classical homotopy theory, the Eilenberg-MacLane functor EM :

D (A b) \to S H

has a nice structural property: after

Q

-localization, the functor

{E M}_{Q} : D (A b)_{Q} \to {S H}_{Q}

is an equivalence. Does the same happen for the motivic Eilenberg-MacLane functor

E M : D M (k) \to S H (k)

? In general, the answer is no, and the reason goes back to Morel's

C - R

dichotomy for

S H (k)

We discuss the case of a characteristic zero field

k

as base. Suppose that

k

admits a real embedding

σ : k \to R

. The embedding

σ

induces a realization functor

R_{R}^{σ} : S H (k) \to S H

which sends the

P^{1}

-suspension spectrum

Σ_{P^{1}}^{\infty} X_{+}

of a smooth

k

-scheme

X

to the infinite suspension spectrum of the real manifold of real points

X (R)

. For an embedding

σ : k \to C

, one has the realization functor

R_{C}^{σ} : S H (k) \to S H

, sending

Σ_{P^{1}}^{\infty} X_{+}

Σ^{\infty} X (C)_{+}

. If we take

X = P^{1}

, the real embedding gives you

Σ S

and the complex embedding yields

Σ^{2} S

, since

P^{1} (R) = S^{1}, P^{1} (C) = S^{2}

. This has the effect that the switch map

τ : P^{1} \land P^{1} \to P^{1} \land P^{1}

induces an automorphism of

S_{k}

that maps to -1 under the real embedding and to +1 under the complex embedding. Thus, if we invert 2 and decompose the motivic sphere spectrum into

\pm 1

eigenfactors with respect to

τ

, we decompose

S H (k) [1 / 2]

into corresponding summands

S H (k)_{\pm}

, with all of

S H (k)_{+}

going to zero under the real embedding and all of

Alternatively, the minus part is

SH (k) [1 / 2, η^{- 1}]

, where

η

is the

P^{1}

-stabilization of the algebraic Hopf map

η : A^{2} ∖ {0} \to P^{1}, η (x, y) = [x : y]

A motivic spectrum

E \in S H (k)

is orientable if

E

has a good theory of Thom classes. For

V \to X

a vector bundle with 0 -section

s_{0} : X \to V

, we have the Thom space

Th (V) := V / (V ∖ s_{0} (X)) \in H_{∙} (k)

(defined as the quotient of representable presheaves). An orientation for

E

consists of giving a class

th (V) \in E^{2 r, r} (Th (V))

for each rank

r

vector bundle

V \to X

over

X \in {S m}_{k}

, satisfying axioms parallel to the notion of a

C

-orientation in topology; a choice of Thom classes defines

E

as an oriented cohomology theory. After inverting 2, all the orientable

E

live in the plus part; this includes motivic cohomology, as well as algebraic

K

-theory and algebraic cobordism. These theories

E

all have the property that

η

induces zero on

E

-cohomology.

Theories that live in the minus part will contrariwise invert

η

(after inverting 2); these include things like Witt theory or cohomology of the sheaf of Witt groups. The real and complex avatars of this are seen by noting that the complex realization of the algebraic Hopf map is the usual Hopf map, which is the 2-torsion element of stable

π_{1}

of the sphere spectrum, while the real realization is the multiplication map

\times 2 : S^{1} \to S^{1}

The analog of the fact that

{E M}_{Q} : D (Q) \to {S H}_{Q}

Theorem 2.4 ([33, THEOREM 16.2.13]). The unit map

S_{k} \to EM (Z (0))

induces an isomorphism

S H (k)_{+ Q} \to DM (k)_{Q}

with inverse the Eilenberg-MacLane functor followed by the plus-projection

D M (k)_{Q} \to S H (k)_{Q} \to S H (k)_{+ Q}

The rational minus part is also a homotopy category of modules over a suitable cohomology theory, namely Witt sheaf cohomology. For a field

F

, we have the Witt ring

W (F)

of virtual non-degenerate quadratic forms, modulo the hyperbolic form. This extends to a sheaf

W

{S m}_{k}

, and the functor

X \mapsto H_{N i s}^{p} (X, W)

is represented in

S H (k)

by a suitable spectrum

EM (W)

. We have

Theorem 2.5 ([3, THEOREM 4.2, COROLLARY 4.4]). The functor

E \mapsto E M (W)_{Q} \land E

induces a natural isomorphism of

S H (k)_{- Q}

with the homotopy category

E M (W)_{Q}

-modules.

From this point of view, one can see the

Z

-graded cohomology theory

X \mapsto ⨁_{n \geq 0} H_{N i s}^{n} (X, W)

as the motivic cohomology for the minus part; this theory picks up information about the real points of schemes. To get the complete theory, one also needs to include twists of

W

by line bundles, an analog of orientation local systems in the topological setting. We will say more about this in Section 4.

2.7. Slice tower and motivic Atiyah-Hirzebruch spectral sequences

The classical Atiyah-Hirzebruch spectral sequence for a spectrum

E \in S H

is the spectral sequence of the Postnikov tower of

E

, and looks like

E_{2}^{p, q} := H^{p} (X, π_{- q} E) \Rightarrow E^{p + q} (X)

This comes by identifying the

q

th layer in the Postnikov tower with the shifted EilenbergMacLane spectrum

Σ^{q} E M (π_{q} (E))

Together with results of Pelaez [99] and Gutierrez-RÃ¶ndigs-Spitzweck, Voevodsky's isomorphism (2.2) has a structural expression, namely, for any

E \in S H (k)

, each slice

s_{q} (E)

has a canonical structure of an

EM (Z (0))

-module. We write corresponding object of

DM (k)

π_{q}^{mot} (E)

, satisfying

s_{q} (E) = Σ_{P^{1}}^{q} EM (π_{q}^{mot} (E)) = S^{2 q, q} \land EM (π_{q}^{mot} (E))

This gives the motivic Atiyah-Hirzebruch spectral sequence

E_{2}^{p, q} (n) := H^{p - q} (X, π_{- q}^{m o t} (E) (n - q)) \Rightarrow E^{p + q, n} (X)

These slices have been explicitly identified in a number of important cases. The first case was algebraic

K

-theory,

K G L \in S H (k)

. Voevodsky [118,119] and Levine [85] show

s_{q} (K G L) = EM (Z (q) [2 q]) = Σ_{P^{1}}^{q} EM (Z (0))

π_{q}^{m o t} (K G L) = Z (0)

corresponding to classical computation for topological

K

-theory,

π_{q}^{s} K U = {\begin{cases} Z & for q even \\ 0 & for q odd. \end{cases}

Using "algebraic Bott periodicity" for KGL:

{K G L}^{a + 2 n, b + n} (X) = {K G L}^{a, b} (X) = K_{2 b - a} (X)

, this yields the Atiyah-Hirzebruch spectral sequence of the Beilinson-Lichtenbaum axiom (v),

E_{2}^{p, q} := H^{p - q} (X, Z (- q)) \Rightarrow K_{- p - q} (X)

There is also a corresponding spectral sequence with

Z / m

-coefficients.

This Atiyah-Hirzebruch spectral sequence for algebraic

K

-theory was first constructed for

X

the spectrum of a field by Bloch and Lichtenbaum [29], by a completely different approach and without recourse to motivic homotopy theory or Voevodsky's slice tower. Their construction was generalized to general

X \in {S m}_{k}

by Friedlander-Suslin [43], also without using the categorical machinery. The rough idea is to give a filtration by codimension of support on

X \times Δ^{*}

(with additional conditions), and then identify the layers with a suitable complex of cycles. Another approach, by Grayson [54], relies on the

K

-theory of exact categories with commuting isomorphisms. For smooth finite-type schemes over a perfect field, all these approaches yield the same spectral sequence (see [85, THEOREM 7.1.1, THEOREM 9.0.3], [44])).

3. MOTIVIC COHOMOLOGY OVER A GENERAL BASE

It is natural to ask if this picture of a good motivic cohomology theory for schemes over a perfect field can be extended to more general base-schemes, not just as an interesting technical question but for a wide range of applications, especially in arithmetic. Over a perfect field, we have a number of different constructions that all lead to the same groups, each of which have their advantages and disadvantages: Bloch's higher Chow groups, the cohomology of a suitable Suslin complex, the morphisms in

D M (k)

, the cohomology theory represented in

S H (k)

EM (Z (0))

, or by

s_{0} S_{k}

, or by

s_{0} K G L

One would expect motivic cohomology to be an absolute theory, like algebraic

K -

theory, that is, its value on a given scheme should not depend on the choice of base-scheme. In terms of a spectrum

H Z_{S} \in S H (S)

that would represent our putative theory, this is the cartesian condition: there should be canonical isomorphisms

H Z_{T} ≅ f^{*} H Z_{S}

for each morphism of schemes

f : T \to S

The identity (2.2) raises the possibility of defining motivic cohomology over a general base-scheme

B

by this formula. One problem here is that the slice filtration has only a limited functoriality: for

f : C \to B

a map of schemes, one does not in general have a natural isomorphism

f^{*} \circ s_{0} ≅ s_{0} \circ f^{*}

. For the cartesian property to hold for a motivic cohomology defined via the slice filtration, one would want the compatibility of the slices
with pullback; this latter is in fact the case for

f : C \to B

is a morphism of separated, finite type schemes over a field

k

of characteristic zero (or assuming resolution of singularities for separated, finite type

k

-schemes), by results of Pelaez [100, COROLLARY 4.3]. This compatibility also holds for arbitrary smooth

f

, but is not known in general.

Another concrete candidate for the motivic Borel-Moore homology is given by the hypercohomology of a version of Bloch's cycle complex, suitably extended to the setting of finite type schemes over a Dedekind domain. This theory is nearly absolute, as it depends only on a good notion of dimension or codimension, which one would have for say equiKrull-dimensional schemes. In general, however, this theory lacks a full functoriality under pullback and also lacks a multiplicative structure.

There is a

P^{1}

-spectrum

{K G L}_{S} \in S H (S)

that represents the so-called homotopy invariant

K

-theory over an arbitrary base and is cartesian, so one could try

s_{0}

KGL as a representing spectrum. Again, the problem is the functoriality of the slice filtration, but perhaps KGL would be easier to handle than the sphere spectrum in this regard.

Over an base-scheme

S

that is noetherian and of finite Krull dimension, CisinskiDÃ©glise [33, Â§11] have followed the program of Voevodsky to define a triangulated category of motives

{D M}_{C D} (S)

, with Tate objects

Z_{S} (n)

, and with a "motives functor"

M : {Sm}_{S} \to {DM}_{C D} (S); X \mapsto M (X) \in {DM}_{C D} (S)

This extends Voevodsky's construction of

D M (k)

for a perfect field

k

1. FROM CLASSICAL TO GEOMETRIC LANGLANDS CORRESPONDENCE

1.1. The geometric Satake

1.2. Tamely ramified local geometric Langlands correspondence

1.3. Global geometric Langlands correspondence

Conjecture 1.3.1. There is a canonical equivalence of ∞ âˆž oo\inftyâˆž-categories

2. FROM GEOMETRIC TO CLASSICAL LANGLANDS PROGRAM

2.1. Categorical arithmetic local Langlands

2.2. Global arithmetic Langlands for function fields

2.3. Geometric realization of Jacquet-Langlands transfer

3. APPLICATIONS TO ARITHMETIC GEOMETRY

3.1. Local models of Shimura varieties

3.2. The congruence relation

3.3. Generic Tate cycles on mod p p p\boldsymbol{p}p fibers of Shimura varieties

3.4. The Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives

ACKNOWLEDGMENTS

FUNDING

REFERENCES

XINWEN ZHU

4. ALGEBRAIC AND COMPLEX GEOMETRY

MARC LEVINE

ABSTRACT

MATHEMATICS SUBJECT CLASSIFICATION 2020

KEYWORDS

1. INTRODUCTION

2. BACKGROUND AND HISTORY

2.1. The conjectures of Beilinson and Lichtenbaum

2.2. Bloch's higher Chow groups and Suslin homology

2.3. Quillen-Lichtenbaum conjectures

2.4. Voevodsky's category DM and modern motivic cohomology

2.5. Motivic homotopy theory

2.6. Motivic cohomology and the rational motivic stable homotopy category

2.7. Slice tower and motivic Atiyah-Hirzebruch spectral sequences

3. MOTIVIC COHOMOLOGY OVER A GENERAL BASE

3.1. Cisinski-DÃ©glise motivic cohomology

Conjecture 1.3.1. There is a canonical equivalence of $\infty$ -categories

3.3. Generic Tate cycles on mod $p$ fibers of Shimura varieties